Palindromic Centered Polygonals

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Palindromic Centered Polygonals
n(n+0) n(n+1) n(n+2) n(n+x) n^2+1 n^2+x n^2–x n^2+(n+1) n^2+(n+x)

Introduction

Palindromic numbers are numbers which read the same from

left to right (forwards) as from the right to left (backwards)

Here are a few random examples : 7, 3113, 44611644

Palindromic Centered Polygonals are defined and calculated by these extraordinary intricate and excruciatingly complex formulae. So, these lines are for experts only

Centered Triangulars {Formula 1} Centered Triangulars {Formula 2}	(3n² + 3n + 2) / 2 (3m² – 3m + 2) / 2	Gen. Form. is Sn² +/– Sn + 2) / 2	Centered Decagonals {Formula 1} Centered Decagonals {Formula 2}	(10n² + 10n + 2) / 2 (10m² – 10m + 2) / 2	5base² + 5base + 1
Centered Squares {Formula 1} Centered Squares {Formula 2}	(4n² + 4n + 2) / 2 (4m² – 4m + 2) / 2	2base² + 2base + 1 OR base² + (base+1)²	Centered 22-gonal numbers [ Because it is in the OEIS ]	(22n² + 22n + 2) / 2 (22m² – 22m + 2) / 2	11base² + 11base + 1
Centered Pentagonals {Formula 1} Centered Pentagonals {Formula 2}	(5n² + 5n + 2) / 2 (5m² – 5m + 2) / 2		Centered 37-gonal numbers [ Because it is in the OEIS ]	(37n² + 37n + 2) / 2 (37m² – 37m + 2) / 2
Centered Hexagonals {Formula 1} Centered Hexagonals {Formula 2}	(6n² + 6n + 2) / 2 (6m² – 6m + 2) / 2	3base² + 3base + 1	Centered 56-gonal numbers [ Because it should be in the OEIS ]	(56n² + 56n + 2) / 2 (56m² – 56m + 2) / 2	28base² + 28base + 1
Centered Heptagonals {Formula 1} Centered Heptagonals {Formula 2}	(7n² + 7n + 2) / 2 (7m² – 7m + 2) / 2		Centered 666-gonal numbers [ Because 666 is a fetish number ]	(666n² + 666n + 2) / 2 (666m² – 666m + 2) / 2	333base² + 333base + 1
Centered Octagonals {Formula 1} Centered Octagonals {Formula 2}	(8n² + 8n + 2) / 2 (8m² – 8m + 2) / 2	4base² + 4base + 1 OR (2*base + 1)²
Centered Nonagonals {Formula 1} Centered Nonagonals {Formula 2}	(9n² + 9n + 2) / 2 (9m² – 9m + 2) / 2		PLAIN TEXT CENTERED

Centered Polygonals	Change of variables	CUDApalin parameters	Base Correction
Centered Triangulars	Odd base n = 2 * m + 1 Even Base n = 2 * m	A B C → 6 9 4 A B C → 6 3 1	base = CUDAbase * 2 + 1 base = CUDAbase * 2
Centered Squares	n.a.	A B C → 2 2 1	base = CUDAbase
Centered Pentagonals	Odd base n = 2 * m + 1 Even Base n = 2 * m	A B C → 10 15 6 A B C → 10 5 1	base = CUDAbase * 2 + 1 base = CUDAbase * 2
Centered Hexagonals	n.a.	A B C → 3 3 1	base = CUDAbase
Centered Heptagonals	Odd base n = 2 * m + 1 Even Base n = 2 * m	A B C → 14 21 8 A B C → 14 7 1	base = CUDAbase * 2 + 1 base = CUDAbase * 2
Centered Octagonals	n.a.	A B C → 4 4 1	base = CUDAbase
Centered Nonagonals	Odd base n = 2 * m + 1 Even Base n = 2 * m	A B C → 18 27 10 A B C → 18 9 1	base = CUDAbase * 2 + 1 base = CUDAbase * 2
Centered Decagonals	n.a.	A B C → 5 5 1	base = CUDAbase
Centered 22_gonals	n.a.	A B C → 11 11 1	base = CUDAbase
Centered 37-gonals	Odd base n = 2 * m + 1 Even Base n = 2 * m	A B C → 74 111 38 A B C → 74 37 1	base = CUDAbase * 2 + 1 base = CUDAbase * 2
Centered 56_gonals	n.a.	A B C → 28 28 1	base = CUDAbase
Centered 666_gonals	n.a.	A B C → 333 333 1	base = CUDAbase

Palindromic Centered Polygonals

To go from Formula 2 = m

Formula 1 = n.

Proof by substitution whereby n equals m + 1

( S*n^2 – S*n + 2 ) / 2 =

( S*(m+1)^2 – S*(m+1) + 2 ) / 2 =

( S*(m^2+2m+1) – S*(m+1) + 2 ) / 2 =

( S*m^2 + S*2m + S – S*m – S + 2 ) / 2 =

( S*m^2 + S*m + S – S + 2 ) / 2 =

( S*m^2 + S*m + 2 ) / 2 [ QED ]

Centered Triangulars of the form [ (3*n² + 3*n + 2 ) / 2 ] can only start or end with a 0, 1, 4, 5, 6 or 9
Even length palindromes of this form are not possible as they are never divisible by 11.

Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.

1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19

4 can be followed by any number : 40, 41, 42, 43, 44, 45, 46, 47, 48 or 49

5 can only be followed by 3 or 8: 53 or 58

6 can be followed by any number : 60, 61, 62, 63, 64, 65, 66, 67, 68 or 69

9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99

There are no palindromic centered triangulars of uneven lengths 3, 11, 13, 21, 31, 45

Centered Squares of the form [ n² + (n + 1)² ] can only start or end with a 1, 3 or 5
Even length palindromes of this form are not possible as they are never divisible by 11.
Also the sums of two consecutive squares

See Sumsquare.htm#jcr2.

1 can only be followed by an even number : 10, 12, 14, 16 or 18

3 can only be followed by 1 : 31

5 can only be followed by an even number : 50, 52, 54, 56 or 58

There are no palindromic centered squares of uneven lengths 5, 9, 37, 39, 43

Centered Pentagonal Numbers of the form [ (5*n² + 5*n + 2) / 2 ] can only start or end with a 1 or 6
Even length palindromes of this form are not possible as they are never divisible by 11.

1 can be followed by any number : 10, 13, 14, 15, 18 or 19

6 can be followed by any number : 60, 61, 62, 65, 66 or 67

There are no palindromic centered pentagonals of uneven lengths 7, 11, 21

Centered Hexagonal Numbers of the form [ 3*n² + 3*n + 1 ] can only start or end with a 1, 7 or 9
Even length palindromes of this form are not possible as they are never divisible by 11.

1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19

7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79

9 can only be followed by 1 or 6: 91 or 96

There are no palindromic centered hexagonals of uneven lengths 5, 11, 17, 19, 27, 47

Centered Heptagonal Numbers of the form [ (7*n² + 7*n + 2) / 2 ] can only start or end with a 1, 2, 3, 6, 7 or 8

1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19

2 can only be followed by 2 or 7: 22 or 27

3 can be followed by any number : 30, 31, 32, 33, 34, 35, 36, 37, 38 or 39

6 can be followed by any number : 60, 61, 62, 63, 64, 65, 66, 67, 68 or 69

7 can only be followed by 4 or 9: 74 or 79

8 can be followed by any number : 80, 81, 82, 83, 84, 85, 86, 87, 88 or 89

There are no palindromic centered heptagonals of lengths 3, 5, 6, 8, 9, 11, 12, 15, 16, 18, 21, 22, 24, 30, 35, 36, 38, 39, 40, 42, 44, 45, 48, 52

Centered Octagonal Numbers of the form [ 4*n² + 4*n + 1 ] can only start or end with a 1, 5 or 9
Also equal to the Odd-based Squares [ (2*n + 1)² ]

1 can only be followed by an even number : 10, 12, 14, 16 or 18

5 can only be followed by 2 : 52

9 can only be followed by an even number : 90, 92, 94, 96 or 98

There are no palindromic centered octagonals of lengths {see palindromic squares}

Centered Nonagonal Numbers of the form [ (9*n² + 9*n + 2) / 2 ] can only start or end with a 0, 1, 3, 5, 6 or 8
Also equal to Every third Triangular Number
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.

1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19

3 can only be followed by 0 or 5: 30 or 35

5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59

6 can be followed by any number : 60, 61, 62, 63, 64, 65, 66, 67, 68 or 69

8 can only be followed by 2 or 7: 82 or 87

There are no palindromic centered nonagonals of lengths {see palindromic triangulars}

Centered Decagonal Numbers of the form [ 5*n² + 5*n + 1 ] can only start or end with a 1

1 can be followed by any number : 10, 11, 13, 15, 16 or 18

There are no palindromic centered decagonals of lengths 4, 5, 10, 12, 28, 29, 32, 38, 52, 54

Centered Icosadigonal Numbers of the form [ 11*n² + 11*n + 1 ] can only start or end with a 1, 3 or 7
Even length palindromes of this form are not possible as they are never divisible by 11.

1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19

3 can be followed by any number : 30, 31, 32, 33, 34, 35, 36, 37, 38 or 39

7 can only be followed by 1 or 6: 71 or 76

There are no palindromic centered icosadigonals of uneven lengths {3 up to 7}, 11, 15, 17, 23, 27, 31, 35, {45 up to 55}

Centered Triacontaheptagonal Numbers of the form [ (37*n² + 37*n + 2) / 2 ] can only start or end with a 1, 2, 3, 6, 7 or 8
Even length palindromes of this form are not possible as they are never divisible by 11.

1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19

2 can only be followed by 1 or 6: 21 or 26

3 can be followed by any number : 30, 31, 32, 33, 34, 35, 36, 37, 38 or 39

6 can be followed by any number : 60, 61, 62, 63, 64, 65, 66, 67, 68 or 69

7 can only be followed by 3 or 8: 73 or 78

8 can be followed by any number : 80, 81, 82, 83, 84, 85, 86, 87, 88 or 89

There are no palindromic centered triacontaheptagonals of uneven lengths 3, 5, 11, {17 up to 23}, 29, 33, 35, {43 up to 55}

Centered Pentacontahexagonal Numbers of the form [ (56*n² + 56*n + 2) / 2 ] can only start or end with a 1, 7, 9

1 can be followed by an even number : 10, 12, 14, 16 or 18

7 can be followed by an odd number : 71, 73, 75, 77 or 79

9 can only be followed by 6: 96

There are no palindromic centered pentacontahexagonals of lengths {2 up to 19}, 21, {23 up to 32}, 34, 35, 37, 38, {40 up to 42}, 45, {47 up to 54}

Centered 666-gonal Numbers of the form [ (666*n² + 666*n + 2) / 2 ] can only start or end with a 1, 7, 9

Even length palindromes of this form are not possible as they are never divisible by 11.

1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19

7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79

9 can only be followed by 4 or 9: 94 or 99

There are no palindromic centered 666-gonals of uneven lengths {3 up to 11}, 15, 17, {21 up to 29}, 33, 37, {43 up to 53}

Two distinct prime factors - a nice semiprime!

CP3{14} = 7573558993381942471 * 10422302211210419163397

CP5{29} = 32218588608327889 * 455383597530039769

CP666{8} = 897356382150563297 * 803921612700156115391

The Table

Index Nr	Basenumber	Length
Index Nr	Palindromic Centered Polygonals	Length

	[CP3] Case X = 3 Palindromic Centered Triangulars
One can find the regular numbers of the form {Formula 2} (3m² – 3m + 2) / 2 at %N (3m² – 3m + 2) / 2. under A005448. The palindromic numbers of the form (3m² – 3m + 2) / 2 are categorised as follows : %N (3m² – 3m + 2) / 2 is a palindrome. under A195903. %N Palindromes of the form (3m² – 3m + 2) / 2. under A162703. Nevertheless the following table will use {Formula 1 = (3n² + 3n + 2) / 2} for its base numbers (m=n+1).

	Index 1 up to 17 were taken from the OEIS (See above) Index 17 up to 30 by Patrick De Geest [ Last one May 15, 2021 ] Index 31 up to 36 by Robert Xiao [ November 11, 2022 ] Index 37 up to 41 by Robert Xiao [ December 6, 2022 ] Index 42 up to 47 by Patrick De Geest [ July 24, 2023 ]
47	2.547.868.070.396.112.441.369.754.748	28
47	9.737.447.556.216.014.074.722.593.161.613.952.274.704.106.126.557.447.379	55
46	1.648.148.920.619.448.676.416.714.726	28
46	4.074.592.296.808.580.601.670.488.533.358.840.761.060.858.086.922.954.704	55
45	180.392.832.671.765.540.424.080.433	27
45	48.812.361.119.015.401.508.234.463.136.443.280.510.451.091.116.321.884	53
44	111.321.898.188.283.945.417.096.679	27
44	18.588.847.524.363.984.553.455.078.187.055.435.548.936.342.574.888.581	53
43	97.600.196.848.996.557.559.318.560	26
43	14.288.697.637.444.316.344.537.051.515.073.544.361.344.473.679.688.241	53
42	95.245.228.486.939.647.972.219.979	26
42	13.607.480.324.294.009.735.726.819.991.862.753.790.049.242.308.470.631	53
41	20.420.461.395.365.543.651.626.149	25
41	625.492.865.399.421.729.116.357.777.753.611.927.124.993.568.294.526	51
40	1.758.243.409.388.476.472.498.593	25
40	4.637.129.829.987.020.513.693.036.303.963.150.207.899.289.217.364	49
39	251.213.024.187.730.896.484.111	24
39	94.661.975.282.318.202.664.878.187.846.620.281.328.257.916.649	47
38	210.314.107.948.093.167.546.810	24
38	66.348.036.003.003.278.504.547.474.540.587.230.030.063.084.366	47
37	168.900.873.407.332.158.114.961	24
37	42.791.257.556.639.465.068.652.325.686.056.493.665.575.219.724	47
36	2.501.246.330.675.987.139.668	22
36	9.384.349.810.080.134.407.354.537.044.310.800.189.434.839	43
35	2.499.975.575.943.076.901.468	22
35	9.374.816.820.467.878.595.887.885.958.787.640.286.184.739	43
34	1.715.263.922.605.217.558.121	22
34	4.413.195.486.286.556.653.769.673.566.556.826.845.913.144	43
33	252.216.016.136.438.237.916	21
33	95.419.378.193.604.110.420.002.401.140.639.187.391.459	41
32	111.395.817.481.145.177.279	21
32	18.613.542.228.438.914.203.230.241.983.482.224.531.681	41
31	96.566.847.292.295.321.404	20
31	13.987.733.993.960.226.370.807.362.206.939.933.778.931	41
30	17.729.422.815.023.433.061	20
30	471.498.650.030.810.150.303.051.018.030.056.894.174	39
29	8.781.786.523.212.541.115	19
29	115.679.661.808.916.116.404.611.619.808.166.976.511	39
28	1.698.666.724.874.603.133	19
28	4.328.202.963.294.315.983.895.134.923.692.028.234	37
27	1.144.251.252.690.582.764	19
27	1.963.966.393.925.951.835.381.595.293.936.693.691	37
26	82.738.939.025.750.800	17
26	Prime! 10.268.598.046.660.363.236.306.664.089.586.201	35
25	17.108.271.682.888.341	17
25	439.039.439.963.278.626.872.369.934.930.934	33
24	10.020.895.012.660.700	17
24	150.627.505.282.152.151.251.282.505.726.051	33
23	213.031.868.452.709	15
23	68.073.865.464.678.787.646.456.837.086	29
22	189.344.689.666.187	15
22	53.777.117.257.177.277.175.271.177.735	29
21	181.002.188.703.853	15
21	49.142.688.473.378.087.337.488.624.194	29
20	170.438.376.135.141	15
20	43.573.860.089.375.957.398.006.837.534	29
19	82.406.037.714.599	14
19	10.186.132.577.729.992.777.523.168.101	29
18	17.476.134.422.426	14
18	458.122.911.526.080.625.119.221.854	27
17	2.036.493.584.525	13
17	6.220.959.179.720.279.719.590.226	25
16	Prime! 196.759.830.167	12
16	58.071.646.151.315.164.617.085	23
15	167.526.192.486	12
15	42.097.537.753.535.773.579.024	23
14	1.702.002.466	10
14	4.345.218.593.958.125.434	19
13	215.198.694	9
13	69.465.717.171.756.496	17
12	17.880.586	8
12	479.573.060.375.974	15
11	Prime! 9.531.979	7
11	136.287.949.782.631	15
10	20.794	5
10	648.616.846	9
9	Prime! 2.069	4
9	6.424.246	7
8	1.821	4
8	4.976.794	7
7	256	3
7	Prime! 98.689	5
6	248	3
6	92.629	5
5	210	3
5	66.466	5
4	Prime! 173	3
4	45.154	5
3	100	3
3	15.151	5
2	1	1
2	4	1
1	0	1
1	1	1

	[CP4] [SUSQ2] Case X = 4 Palindromic Centered Squares
One can find the regular numbers of the form n² + (n+1)² at %N n² + (n+1)². under A001844. The palindromic numbers of the form n² + (n+1)² are categorised as follows : %N n² + (n+1)² is a palindrome. under A027571. %N Palindromes of the form n² + (n+1)². under A027572. The palindromic primes of the form n² + (n+1)² are categorised as follows : %N n² + (n+1)² is a palindromic prime. under A050236. %N Palindromic primes of the form n² + (n+1)². under A050239.

	Index 47 up to 52 by Patrick De Geest [ Last one June 16, 2021 ] Index 53 up to 59 by Robert Xiao [ December 6, 2022 ] Index 60 up to 63 by Patrick De Geest [ June 13, 2023 ]
63	1.262.413.288.821.833.950.379.107.737	28
63	3.187.374.623.587.918.287.303.295.065.605.923.037.828.197.853.264.737.813	55
62	781.149.338.627.822.004.402.338.114	27
62	1.220.388.578.477.367.461.271.198.953.598.911.721.647.637.748.758.830.221	55
61	722.342.892.165.927.171.629.878.924	27
61	1.043.558.507.725.272.580.121.092.036.302.901.210.852.725.277.058.553.401	55
60	125.845.220.398.502.946.907.933.882	27
60	31.674.038.994.295.564.414.718.234.143.281.741.446.559.249.983.047.613	53
59	12.485.050.866.969.785.154.347.907	26
59	311.752.990.301.645.967.838.382.111.283.838.769.546.103.099.257.113	51
58	12.463.239.245.524.015.944.014.602	26
58	310.664.664.982.340.084.364.343.252.343.463.480.043.289.466.466.013	51
57	9.137.327.500.983.204.356.513.194	25
57	166.981.507.720.447.940.821.484.646.484.128.049.744.027.705.189.661	51
56	739.558.350.036.969.165.689.450	24
56	1.093.893.106.218.808.420.715.399.935.170.248.088.126.013.983.901	49
55	79.608.488.039.503.229.466.585	23
55	12.675.022.735.871.457.478.928.682.987.475.417.853.722.057.621	47
54	72.971.763.254.756.896.408.324	23
54	10.649.756.465.016.577.599.097.379.099.577.561.056.465.794.601	47
53	9.057.485.682.660.759.666.330	22
53	164.076.093.783.209.295.116.999.611.592.902.387.390.670.461	45
52	89.940.695.078.362.814.444	20
52	16.178.657.262.358.073.985.458.937.085.326.275.687.161	41
51	84.753.839.053.503.067.854	20
51	14.366.426.468.614.203.601.510.630.241.686.462.466.341	41
50	84.284.587.587.769.426.720	20
50	14.207.783.409.680.752.220.402.225.708.690.438.770.241	41
49	124.566.258.570.164.697	18
49	31.033.505.548.338.259.895.283.384.550.533.013	35
48	91.424.589.460.359.855	17
48	16.716.911.115.990.684.748.609.951.111.961.761	35
47	89.663.751.709.213.505	17
47	16.079.176.741.142.975.657.924.114.767.197.061	35
46	72.421.835.667.809.200	17
46	10.489.844.562.990.321.812.309.926.544.898.401	35
1 - 45	See full list at Sumsquare.htm

	[CP5] Case X = 5 Palindromic Centered Pentagonals
One can find the regular numbers of the form (5n² + 5n + 2) / 2 at %N (5n² + 5n + 2) / 2. under A005891. The palindromic numbers of the form (5n² + 5n + 2) / 2 are categorised as follows : %N (5n² + 5n + 2) / 2 is a palindrome. under A??????. %N Palindromes of the form (5n² + 5n + 2) / 2. under A??????.

	Index 1 up to 38 by Patrick De Geest [ not dated ] Index 39 up to 43 by Robert Xiao [ November 11, 2022 ] Index 44 up to 54 by Robert Xiao [ December 6, 2022 ] Index 55 up to 56 by Patrick De Geest [ July 24, 2023 ]
56	1.627.604.670.871.899.611.753.825.337	28
56	6.622.742.411.610.061.651.209.087.380.837.809.021.561.600.161.142.472.266	55
55	72.283.120.766.097.573.993.947.171	26
55	13.062.123.869.215.615.830.830.545.554.503.803.851.651.296.832.126.031	53
54	8.554.792.163.507.163.450.358.271	25
54	182.961.172.402.008.935.975.567.939.765.579.539.800.204.271.169.281	51
53	8.524.775.014.165.545.823.091.208	25
53	181.679.472.605.352.954.971.868.505.868.179.459.253.506.274.976.181	51
52	7.655.591.981.451.737.262.290.927	25
52	146.520.221.466.170.341.716.576.757.675.617.143.071.664.122.025.641	51
51	7.273.159.187.548.388.682.985.043	25
51	132.247.111.418.548.843.361.749.323.947.163.348.845.814.111.742.231	51
50	6.342.474.339.344.580.849.484.624	25
50	100.567.451.863.111.193.280.632.101.236.082.391.111.368.154.765.001	51
49	1.562.793.226.754.366.380.998.802	25
49	6.105.806.673.973.311.042.157.989.897.512.401.133.793.766.085.016	49
48	867.314.050.987.862.637.736.576	24
48	1.880.584.157.602.441.978.334.646.464.338.791.442.067.514.850.881	49
47	164.700.324.803.040.506.984.874	24
47	67.815.492.475.567.600.039.850.605.893.000.676.557.429.451.876	47
46	161.336.156.493.053.137.848.866	24
46	65.073.388.479.877.330.429.124.642.192.403.377.897.488.337.056	47
45	76.573.638.714.774.500.006.352	23
45	14.658.805.365.052.030.352.843.634.825.303.025.056.350.885.641	47
44	15.529.695.907.925.431.315.486	23
44	602.928.637.481.589.716.186.242.681.617.985.184.736.829.206	45
43	744.357.542.421.929.220.588	21
43	1.385.170.377.401.035.398.121.218.935.301.047.730.715.831	43
42	89.191.300.798.092.459.892	20
42	19.887.720.345.139.521.682.428.612.593.154.302.778.891	41
41	78.502.340.637.995.800.155	20
41	15.406.543.714.109.817.129.492.171.890.141.734.560.451	41
40	78.486.443.999.610.951.780	20
40	15.400.304.729.260.164.943.534.946.106.292.740.300.451	41
39	77.929.442.346.456.135.660	20
39	Prime! 15.182.494.961.074.076.947.574.967.047.016.949.428.151	41
38	8.545.229.924.052.920.928	19
38	182.552.386.137.323.721.949.127.323.731.683.255.281	39
37	7.411.900.058.658.854.468	19
37	Prime! 137.340.656.198.867.825.777.528.768.891.656.043.731	39
36	6.548.959.543.466.373.944	19
36	107.222.177.754.898.242.595.242.898.457.771.222.701	39
35	6.455.494.583.580.369.039	19
35	104.183.525.796.588.705.676.507.885.697.525.381.401	39
34	6.334.531.182.684.034.799	19
34	100.315.713.260.990.991.646.199.099.062.317.513.001	39
33	766.048.834.768.223.792	18
33	1.467.077.043.124.383.593.953.834.213.407.707.641	37
32	644.147.472.408.716.104	18
32	1.037.314.915.526.344.187.814.436.255.194.137.301	37
31	158.641.107.117.468.129	18
31	62.917.502.168.639.993.039.993.686.120.571.926	35
30	156.906.204.779.292.757	18
30	61.548.892.745.603.383.438.330.654.729.884.516	35
29	76.607.615.255.405.807	17
29	14.671.816.787.800.711.511.700.878.761.817.641	35
28	74.745.660.814.331.483	17
28	13.967.284.526.427.722.622.772.462.548.276.931	35
27	65.323.181.999.789.040	17
27	10.667.795.266.443.907.270.934.466.259.776.601	35
26	7.963.013.978.107.115	16
26	158.523.979.038.823.272.328.830.979.325.851	33
25	1.615.286.421.727.426	16
25	6.522.875.560.542.483.842.450.655.782.256	31
24	647.673.577.893.760	15
24	1.048.702.658.754.262.624.578.562.078.401	31
23	79.049.055.962.164	14
23	15.621.883.121.273.537.212.138.812.651	29
22	15.846.566.505.009	14
22	627.784.174.994.222.499.471.487.726	27
21	7.512.476.533.592	13
21	141.093.259.169.444.961.952.390.141	27
20	7.435.245.752.988	13
20	138.207.198.518.333.815.891.702.831	27
19	1.553.657.288.041	13
19	6.034.627.421.711.171.247.264.306	25
18	793.717.097.724	12
18	Prime! 1.574.967.078.050.508.707.694.751	25
17	65.751.464.704	11
17	10.808.137.776.967.773.180.801	23
16	1.559.881.233	10
16	6.083.073.656.563.703.806	19
15	778.794.860	9
15	1.516.303.586.853.036.151	19
14	76.591.487	8
14	14.665.639.893.656.641	17
13	15.759.110	8
13	620.873.909.378.026	15
12	15.534.441	8
12	603.297.181.792.306	15
11	Prime! 8.528.983	7
11	181.858.898.858.181	15
10	7.834.140	7
10	153.434.393.434.351	15
9	780.699	6
9	1.523.729.273.251	13
8	737.003	6
8	1.357.935.397.531	13
7	8.787	4
7	193.050.391	9
6	6.464	4
6	104.474.401	9
5	64	2
5	10401	5
4	8	1
4	Prime! 181	3
3	Prime! 7	1
3	141	3
2	1	1
2	6	1
1	0	1
1	1	1

	[CP6] Case X = 6 Palindromic Centered Hexagonals
One can find the regular numbers of the form 3n² + 3n + 1 at %N 3n² + 3n + 1. under A003215. The palindromic numbers of the form 3n² + 3n + 1 are categorised as follows : %N 3n² + 3n + 1 is a palindrome. under A??????. %N Palindromes of the form 3n² + 3n + 1. under A??????.

	Index 1 up to 24 by Patrick De Geest [ not dated ] Index 25 and 26 by Robert Xiao [ November 11, 2022 ] Index 27 up to 30 by Robert Xiao [ December 6, 2022 ] Index 31 up to 37 by Patrick De Geest [ July 24, 2023 ]
37	1.541.023.416.503.386.232.028.816.616	28
37	7.124.259.510.635.306.993.850.686.652.566.860.583.996.035.360.159.524.217	55
36	633.910.875.493.346.320.800.755.115	27
36	1.205.528.994.206.222.463.111.196.032.306.911.113.642.226.024.998.255.021	55
35	625.893.836.737.195.960.285.700.329	27
35	1.175.229.284.596.823.134.431.897.462.647.981.344.313.286.954.829.225.711	55
34	75.803.178.161.188.627.109.752.865	26
34	17.238.365.458.010.713.232.202.652.525.620.223.231.701.085.456.383.271	53
33	73.459.781.571.385.652.799.730.320	26
33	16.189.018.525.547.073.506.706.049.894.060.760.537.074.552.581.098.161	53
32	68.841.381.773.920.977.048.254.680	26
32	14.217.407.533.628.217.866.087.971.217.978.066.871.282.633.570.471.241	53
31	58.062.671.575.249.270.307.253.700	26
31	10.113.821.491.365.778.741.776.012.121.067.714.787.756.319.412.831.101	53
30	7.750.385.680.945.922.134.909.760	25
30	180.205.434.610.234.755.415.482.121.284.514.557.432.016.434.502.081	51
29	1.746.738.323.854.645.791.805.222	25
29	9.153.284.316.067.612.332.012.427.242.102.332.167.606.134.823.519	49
28	636.964.368.671.280.296.554.559	24
28	1.217.170.820.870.408.052.426.217.126.242.508.040.780.280.717.121	49
27	17.506.939.206.985.888.982.942	23
27	919.478.761.191.299.122.240.353.042.221.992.191.167.874.919	45
26	688.075.137.861.408.627.680	21
26	1.420.342.186.028.989.466.251.526.649.898.206.812.430.241	43
25	178.985.252.440.612.766.632	21
25	96.107.161.773.689.635.003.230.053.698.637.716.170.169	41
24	17.422.294.995.888.404.722	20
24	910.609.088.771.274.444.979.444.472.177.880.906.019	39
23	15.426.989.438.535.955.891	20
23	713.976.009.410.099.782.797.287.990.014.900.679.317	39
22	7.781.806.191.361.855.164	19
22	181.669.522.799.753.105.999.501.357.997.225.966.181	39
21	731.038.731.693.350.995	18
21	1.603.252.881.707.469.675.769.647.071.882.523.061	37
20	179.258.981.327.928.412	18
20	96.401.347.160.179.761.716.797.106.174.310.469	35
19	153.205.888.386.457.273	18
19	70.416.132.708.850.811.011.805.880.723.161.407	35
18	16.096.354.798.508.231	17
18	777.277.913.398.376.909.673.893.319.772.777	33
17	7.060.467.049.615.580	16
17	149.550.584.876.122.020.221.678.485.055.941	33
16	671.640.175.743.589	15
16	1.353.301.577.018.639.368.107.751.033.531	31
15	160.017.597.111.658	15
15	76.816.894.156.167.176.165.149.861.867	29
14	79.625.800.994.005	14
14	19.020.804.551.810.901.815.540.802.091	29
13	583.937.096.775	12
13	1.022.947.598.971.798.957.492.201	25
12	72.356.279.625	11
12	15.706.293.603.730.639.260.751	23
11	16.208.835.478	11
11	788.179.042.707.240.971.887	21
10	6.617.514	7
10	131.374.494.473.131	15
9	1.750.617	7
9	9.193.984.893.919	13
8	7.760	4
8	180.676.081	9
7	7.479	4
7	167.828.761	9
6	805	3
6	1.946.491	7
5	629	3
5	1.188.811	7
4	600	3
4	1.081.801	7
3	Prime! 17	2
3	Prime! 919	3
2	1	1
2	Prime! 7	1
1	0	1
1	1	1

	[CP7] Case X = 7 Palindromic Centered Heptagonals
One can find the regular numbers of the form (7m² – 7m + 2) / 2) at %N (7m² – 7m + 2) / 2). under A069099. The palindromic numbers of the form (7m² – 7m + 2) / 2) are categorised as follows : %N (7m² – 7m + 2) / 2) is a palindrome. under A??????. %N Palindromes of the form (7m² – 7m + 2) / 2). under A??????. Nevertheless the following table will use {Formula 1 = (7n² + 7n + 2) / 2} for its base numbers (m=n+1).

	Index 1 up to 35 by Patrick De Geest [ not dated ] Index 36 and 37 by Robert Xiao [ November 11, 2022 ] Index 38 up to 46 by Robert Xiao [ December 6, 2022 ] Index 47 up to 51 by Patrick De Geest [ July 24, 2023 ]
51	705.907.453.935.528.503.680.938.020	27
51	1.744.068.667.326.091.037.259.432.113.112.349.527.301.906.237.668.604.471	55
50	499.536.941.655.414.629.450.678.741	27
50	873.380.046.274.557.918.580.573.160.061.375.085.819.755.472.640.083.378	54
49	93.078.429.352.238.483.471.306.091	26
49	30.322.579.037.378.776.658.213.285.358.231.285.667.787.373.097.522.303	53
48	79.408.299.232.608.570.254.049.922	26
48	220.69.872.954.554.259.966.246.329.792.364.266.995.245.545.927.896.022	53
47	64.648.550.821.159.124.341.030.815	26
47	14.628.022.931.465.979.315.479.510.301.597.451.397.956.413.922.082.641	53
46	5.853.245.362.233.523.747.423.035	25
46	Prime! 119.911.684.446.778.891.194.836.060.638.491.198.877.644.486.119.911	51
45	2.977.659.166.374.072.158.959.736	25
45	31.032.589.388.820.370.196.592.833.829.569.107.302.888.398.523.013	50
44	2.507.142.065.529.384.342.466.877	25
44	22.000.164.678.614.317.063.338.388.383.336.071.341.687.646.100.022	50
43	2.226.763.081.260.708.292.853.795	25
43	17.354.658.370.229.893.173.799.466.499.737.139.892.207.385.645.371	50
42	1.592.207.082.821.695.791.623.161	25
42	8.872.931.881.056.510.547.067.102.017.607.450.156.501.881.392.788	49
41	742.924.722.381.444.645.846.059	24
41	1.931.780.000.939.413.096.824.255.524.286.903.149.390.000.871.391	49
40	135.419.664.346.965.483.614.570	24
40	64.184.699.221.456.780.896.126.162.169.808.765.412.299.648.146	47
39	97.560.683.517.514.878.937.848	23
39	33.313.304.389.416.447.800.247.474.200.874.461.498.340.331.333	47
38	23.220.890.900.927.654.224.655	23
38	1.887.234.209.814.746.537.808.558.087.356.474.189.024.327.881	46
37	1.339.381.461.586.827.010.274	22
37	6.278.799.448.748.627.353.335.333.537.268.478.449.978.726	43
36	70.569.029.623.031.986.020	20
36	17.429.957.796.777.280.884.448.808.277.769.775.992.471	41
35	932.758.758.091.321.916	18
35	3.045.136.152.786.228.195.918.226.872.516.315.403	37
34	566.388.600.619.580.460	18
34	Prime! 1.122.786.164.191.323.168.613.231.914.616.872.211	37
33	43.142.652.872.006.569	17
33	6.514.509.738.920.598.448.950.298.379.054.156	34
32	22.442.486.416.824.779	17
32	1.762.828.187.992.776.556.772.997.818.282.671	34
31	8.028.245.530.124.522	16
31	225.584.542.021.875.313.578.120.245.485.522	33
30	2.266.182.015.260.979	16
30	17.974.533.242.023.100.132.024.233.547.971	32
29	1.035.206.065.266.856	16
29	3.750.780.591.478.505.058.741.950.870.573	31
28	733.722.882.837.280	15
28	1.884.222.440.796.673.766.970.442.224.881	31
27	709.830.923.358.779	15
27	1.763.509.789.147.321.237.419.879.053.671	31
26	137.171.486.712.369	15
26	65.856.058.684.086.168.048.685.065.856	29
25	136.845.415.594.294	15
25	65.543.337.192.113.131.129.173.334.556	29
24	70.267.104.137.195	14
24	17.281.130.733.396.169.333.703.118.271	29
23	30.771.414.144.248	14
23	3.314.079.749.528.998.259.479.704.133	28
22	Prime! 20.397.301.735.159	14
22	1.456.174.713.262.992.623.174.716.541	28
21	14.056.310.919.254	14
21	691.529.568.305.636.503.865.925.196	27
20	5.525.124.851.024	13
20	106.844.516.167.929.761.615.448.601	27
19	2.308.275.787.919	13
19	18.648.479.895.833.859.897.484.681	26
18	1.875.549.816.879	13
18	12.311.904.904.588.540.940.911.321	26
17	888.606.676.062	12
17	2.763.676.386.599.956.836.763.672	25
16	639.152.410.415	12
16	1.429.805.313.089.803.135.089.241	25
15	155.707.617.281	12
15	84.857.017.278.187.271.075.848	23
14	66.854.165.724	11
14	15.643.178.161.516.187.134.651	23
13	3.369.744.496	10
13	39.743.122.900.922.134.793	20
12	886.441.782	9
12	2.750.226.618.166.220.572	19
11	648.289.959	9
11	1.470.979.550.559.790.741	19
10	94.660.336	8
10	31.362.027.572.026.313	17
9	3.313.923	7
9	38.437.311.373.483	14
8	1.390.849	7
8	6.770.618.160.776	13
7	Prime! 798.757	6
7	2.233.047.403.322	13
6	32.436	5
6	3.682.442.863	10
5	740	3
5	1.919.191	7
4	Prime! 19	2
4	1.331	4
3	Prime! 2	1
3	22	2
2	1	1
2	8	1
1	0	1
1	1	1

	[CP8] Case X = 8 Palindromic Centered Octagonals
One can find the regular numbers of the form 4n² + 4n + 1 at %N (2n + 1)². under A016754. The palindromic numbers of the form (2n + 1)² are categorised as follows : %N (2n + 1)² is a palindrome. under A??????. %N Palindromes of the form (2n + 1)². under A??????.
all	This is a subset of the palindromic squares namely every palindromic square whose base is “odd” Base_octa becomes ( base b_square – 1 divided by two ). Let me list them up to length 13.	!
38	1.534.653	7
38	9.420.645.460.249	13
37	1.147.337	7
37	5.265.533.355.625	13
36	635.434	6
36	1.615.108.015.161	13
35	555.555	6
35	1.234.567.654.321	13
34	555.055	6
34	1.232.346.432.321	13
33	554.555	6
33	1.230.127.210.321	13
32	551.005	6
32	1.214.428.244.121	13
31	550.505	6
31	1.212.225.222.121	13
30	550.005	6
30	1.210.024.200.121	13
29	521.075	6
29	1.086.078.706.801	13
28	506.050	6
28	1.024.348.434.201	13
27	505.550	6
27	1.022.325.232.201	13
26	505.050	6
26	1.020.304.030.201	13
25	501.000	6
25	1.004.006.004.001	13
24	500.500	6
24	1.002.003.002.001	13
23	500.000	6
23	1.000.002.000.001	13
22	55.555	5
22	12.345.654.321	11
21	55.005	5
21	12.102.420.121	11
20	50.550	5
20	10.221.412.201	11
19	50.000	5
19	10.000.200.001	11
18	15.346	5
18	942.060.249	9
17	11.432	5
17	522.808.225	9
16	5.605	4
16	125.686.521	9
15	5.555	4
15	123.454.321	9
14	5.505	4
14	121.242.121	9
13	5.100	4
13	104.060.401	9
12	5.050	4
12	102.030.201	9
11	5.000	4
11	100.020.001	9
10	1.142	4
10	5.221.225	7
9	555	3
9	1.234.321	7
8	500	3
8	1.002.001	7
7	153	3
7	94.249	5
6	60	2
6	14.641	5
5	55	2
5	12.321	5
4	50	2
4	10.201	5
3	Prime! 5	1
3	121	3
2	1	1
2	9	1
1	0	1
1	1	1

	[CP9] Case X = 9 Palindromic Centered Nonagonals
One can find the regular numbers of the form (9m² – 9m + 2) / 2 at %N (9m² – 9m +2) / 2. under A060544. The palindromic numbers of the form (9m² – 9m + 2) / 2 are categorised as follows : %N (9m² – 9m +2) / 2 is a palindrome. under A??????. %N Palindromes of the form (9m² – 9m +2) / 2. under A??????. Nevertheless the following table will use {Formula 1 = (9n² + 9n + 2) / 2} for its base numbers (m=n+1).
all	This is a subset of the palindromic triangulars namely every palindromic triangular with base b_nona equal to (b_tria – 1) divisible by 3. Let me list them up to length 58.	!
55	12.032.570.833.093.463.942.873.712.365	29
55	651.522.423.840.351.916.117.173.996.292.699.371.711.619.153.048.324.225.156	57
54	10.990.762.874.808.839.860.648.779.136	29
54	543.585.908.566.243.233.447.813.416.222.614.318.744.332.342.665.809.585.345	57
53	1.390.507.544.775.228.373.474.641.062	28
53	8.700.800.544.345.751.829.458.446.650.566.448.549.281.575.434.450.080.078	55
52	570.774.037.557.662.643.334.884.895	27
52	1.466.023.508.774.442.385.882.534.859.584.352.885.832.444.778.053.206.641	55
51	11.641.949.341.786.113.114.882.865	26
51	609.907.430.145.213.505.805.139.181.931.508.505.312.541.034.709.906	51
50	1.093.980.277.256.091.349.948.548	25
50	5.385.567.811.613.915.254.381.275.721.834.525.193.161.187.655.835	49
49	511.682.759.102.703.872.734.019	24
49	1.178.186.606.833.300.573.192.218.122.913.750.033.386.066.818.711	49
48	38.286.913.835.022.303.780.765	23
48	6.596.494.969.546.900.318.903.773.098.130.096.459.694.946.956	46
47	4.723.477.919.141.722.887.999	22
47	100.400.596.436.787.391.913.333.319.193.787.634.695.004.001	45
46	258.593.922.027.612.151.867	21
46	300.918.674.293.302.389.819.918.983.203.392.476.819.003	42
45	111.156.781.896.700.878.028	21
45	55.601.235.727.338.276.212.021.267.283.372.753.210.655	41
44	11.789.941.359.940.700.950	20
44	625.512.227.718.781.732.353.237.187.817.722.215.526	39
43	3.888.611.090.993.192.105	19
43	68.045.832.976.478.686.977.968.687.467.923.854.086	38
42	1.217.414.068.855.875.345	19
42	6.669.436.567.716.980.985.890.896.177.656.349.666	37
41	Prime! 1.212.527.593.913.419.529	19
41	6.616.004.247.006.598.875.788.956.007.424.006.166	37
40	1.079.177.623.768.796.591	19
40	5.240.809.546.394.698.286.828.964.936.459.080.425	37
39	258.509.259.706.890.107	18
39	300.721.668.093.919.607.706.919.390.866.127.003	36
38	117.483.227.172.259.625	18
38	62.110.389.000.639.430.803.493.600.098.301.126	35
37	37.513.595.580.544.394	17
37	6.332.714.340.212.879.669.782.120.434.172.336	34
36	36.033.401.143.279.976	17
36	5.842.826.990.786.388.228.836.870.996.282.485	34
35	26.119.645.131.833.612	17
35	3.070.061.378.158.136.996.318.518.731.600.703	34
34	1.354.087.421.826.197	16
34	8.250.987.356.765.633.365.676.537.890.528	31
33	120.928.470.687.909	15
33	65.806.627.603.124.642.130.672.660.856	29
32	36.412.576.729.763	14
32	5.966.440.848.454.114.548.480.446.695	28
31	13.933.969.125.857	14
31	873.699.730.201.575.102.037.996.378	27
30	3.621.907.975.011	13
30	59.031.978.207.533.570.287.913.095	26
29	1.684.040.145.439	13
29	12.761.960.451.533.515.406.916.721	26
28	135.090.111.242	12
28	82.122.021.699.799.612.022.128	23
27	Prime! 88.814.814.887	11
27	35.496.321.045.754.012.369.453	23
26	Prime! 82.214.815.547	11
26	30.416.741.529.792.514.761.403	23
25	20.915.965.579	11
25	1.968.649.272.552.729.468.691	22
24	8.898.308.552	10
24	356.309.527.929.725.903.653	21
23	Prime! 8.296.010.227	10
23	309.707.035.626.530.707.903	21
22	5.842.580.300	10
22	153.610.850.555.058.016.351	21
21	1.837.866.924	10
21	15.199.896.744.769.899.151	20
20	338.236.268	9
20	514.816.979.979.618.415	18
19	122.835.250	9
19	67.898.244.444.289.876	17
18	49.907.019	8
18	11.208.197.679.180.211	17
17	19.669.939	8
17	1.741.079.339.701.471	16
16	12.326.369	8
16	683.727.232.727.386	15
15	12.309.969	8
15	681.909.070.909.186	15
14	11.890.130	8
14	636.188.414.881.636	15
13	11.568.390	8
13	602.224.464.422.206	15
12	10.950.323	8
12	539.593.131.395.935	15
11	4.270.797	7
11	82.078.700.787.028	14
10	59.719	5
10	16.048.884.061	11
9	55.684	5
9	13.953.435.931	11
8	19.055	5
8	1.634.004.361	10
7	16.760	5
7	1.264.114.621	10
6	Prime! 887	3
6	3.544.453	7
5	370	3
5	617.716	6
4	36	2
4	5.995	4
3	Prime! 11	2
3	595	3
2	Prime! 3	1
2	55	2
1	0	1
1	1	1

	[CP10] Case X = 10 Palindromic Centered Decagonals
One can find the regular numbers of the form 5n² + 5n + 1 at %N 5n² + 5n + 1. under A062786. The palindromic numbers of the form 5n² + 5n + 1 are categorised as follows : %N 5n² + 5n + 1 is a palindrome. under A??????. %N Palindromes of the form 5n² + 5n + 1. under A??????. The palindromic primes of the form 5n² + 5n + 1 are categorised as follows : %N 5n² + 5n + 1 is a palindromic prime. under A134463. %N Palindromic primes of the form 5n² + 5n + 1. under A134462.

	Index 1 up to 86 by Patrick De Geest [ not dated ] Index 87 up to 96 by Robert Xiao [ November 11, 2022 ] Index 97 up to 110 by Robert Xiao [ December 6, 2022 ] Index 111 up to 115 by Patrick De Geest [ July 24, 2023 ]
115	573.577.733.632.465.461.995.343.331	27
115	1.644.957.082.594.777.505.920.929.649.469.290.295.057.774.952.807.594.461	55
114	488.166.204.429.814.852.576.574.261	27
114	1.191.531.215.737.058.930.974.962.803.082.694.790.398.507.375.121.351.911	55
113	473.742.410.875.559.518.790.395.193	27
113	1.122.159.359.310.937.269.135.043.392.933.405.319.627.390.139.539.512.211	55
112	45.479.861.861.580.682.317.168.364	26
112	10.342.089.174.742.305.432.306.112.021.160.323.450.324.747.198.024.301	53
111	45.253.928.773.533.389.735.642.959	26
111	10.239.590.347.200.166.238.696.780.408.769.683.266.100.274.309.593.201	53
110	5.714.937.055.687.488.063.188.191	25
110	163.302.527.752.349.875.199.206.969.602.991.578.943.257.725.203.361	51
109	4.874.308.420.162.336.114.068.153	25
109	118.794.412.874.327.244.876.856.575.658.678.442.723.478.214.497.811	51
108	4.807.766.703.909.756.877.335.173	25
108	115.573.103.396.116.439.275.911.313.119.572.934.611.693.301.375.511	51
107	1.764.480.851.877.385.668.630.789	25
107	15.566.963.383.209.723.121.257.622.675.212.132.790.238.336.966.551	50
106	1.453.112.726.116.570.815.725.100	25
106	10.557.682.974.009.660.738.171.388.317.183.706.690.047.928.675.501	50
105	1.426.857.413.135.825.667.820.979	25
105	10.179.610.387.103.301.455.152.644.625.155.410.330.178.301.697.101	50
104	554.905.320.418.079.794.050.970	24
104	1.539.599.573.141.459.019.907.832.387.099.109.541.413.759.959.351	49
103	553.065.221.503.128.132.987.350	24
103	1.529.405.696.181.520.926.885.353.535.886.290.251.816.965.049.251	49
102	181.939.900.170.799.732.386.071	24
102	165.510.636.370.802.862.449.348.843.944.268.208.073.636.015.561	48
101	47.715.025.388.664.598.050.421	23
101	11.383.618.239.204.535.881.208.980.218.853.540.293.281.638.311	47
100	46.546.080.186.424.215.021.464	23
100	10.832.687.903.605.164.437.032.823.073.446.150.630.978.623.801	47
99	17.716.206.023.340.562.407.030	23
99	1.569.319.779.307.242.120.311.991.130.212.427.039.779.139.651	46
98	14.780.401.775.203.197.081.444	23
98	1.092.301.383.182.149.098.157.777.518.909.412.813.831.032.901	46
97	4.530.524.161.109.363.244.359	22
97	102.628.245.871.978.497.814.020.418.794.879.178.542.826.201	45
96	1.804.209.648.549.004.015.276	22
96	16.275.862.279.586.602.933.033.033.920.668.597.226.857.261	44
95	577.173.877.136.554.056.636	21
95	1.665.648.422.244.209.987.982.897.899.024.422.248.465.661	43
94	520.893.003.920.459.673.942	21
94	1.356.647.607.666.400.086.239.326.800.046.667.067.466.531	43
93	510.808.240.259.791.104.642	21
93	1.304.625.291.586.522.369.638.369.632.256.851.925.264.031	43
92	482.568.843.543.085.582.566	21
92	1.164.363.443.792.555.058.765.678.505.552.973.443.634.611	43
91	175.640.382.470.360.338.265	21
91	154.247.719.771.672.316.012.210.613.276.177.917.742.451	42
90	144.813.077.840.713.266.719	21
90	104.854.137.568.502.398.817.718.893.205.865.731.458.401	42
89	45.716.218.669.549.868.480	20
89	10.449.863.247.210.499.734.743.799.401.274.236.894.401	41
88	45.348.805.706.682.784.040	20
88	10.282.570.895.112.325.200.100.252.321.159.807.528.201	41
87	18.415.328.375.174.250.056	20
87	1.695.621.595.827.489.423.223.249.847.285.951.265.961	40
86	6.036.281.167.110.772.127	19
86	182.183.451.642.080.926.515.629.080.246.154.381.281	39
85	6.033.764.984.134.155.592	19
85	182.031.599.418.817.234.444.432.718.814.995.130.281	39
84	5.760.242.508.141.202.328	19
84	Prime! 165.901.968.762.984.246.868.642.489.267.869.109.561	39
83	4.611.483.393.881.208.304	19
83	106.328.895.460.210.736.868.637.012.064.598.823.601	39
82	612.734.698.031.971.332	18
82	1.877.219.050.861.655.467.645.561.680.509.127.781	37
81	580.453.550.554.201.916	18
81	1.684.631.621.754.897.200.027.984.571.261.364.861	37
80	560.645.404.450.209.850	18
80	1.571.616.347.655.696.916.196.965.567.436.161.751	37
79	559.469.308.539.469.894	18
79	1.565.029.535.988.162.807.082.618.895.359.205.651	37
78	473.572.348.147.542.593	18
78	1.121.353.844.649.886.444.446.889.464.483.531.211	37
77	460.593.464.512.444.279	18
77	1.060.731.697.757.881.339.331.887.577.961.370.601	37
76	459.728.007.205.710.324	18
76	1.056.749.203.046.668.220.228.666.403.029.476.501	37
75	152.957.448.255.024.433	18
75	116.979.904.883.442.385.583.244.388.409.979.611	36
74	144.078.685.966.944.235	18
74	103.793.338.749.806.668.866.608.947.833.397.301	36
73	142.171.056.703.991.995	18
73	101.063.046.821.648.536.635.846.128.640.360.101	36
72	56.473.953.204.117.765	17
72	15.946.536.952.504.416.161.440.525.963.564.951	35
71	52.002.994.083.007.622	17
71	13.521.556.967.986.628.982.668.976.965.512.531	35
70	45.723.888.363.764.680	17
70	10.453.369.835.510.075.757.001.553.896.335.401	35
69	19.081.039.958.424.167	17
69	1.820.430.429.474.898.778.984.749.240.340.281	34
68	18.126.473.120.621.468	17
68	1.642.845.138.963.062.992.603.698.315.482.461	34
67	17.423.506.104.996.594	17
67	1.517.892.824.954.267.997.624.594.282.987.151	34
66	17.404.751.100.767.930	17
66	1.514.626.804.398.412.442.148.934.086.264.151	34
65	16.149.368.108.154.557	17
65	1.304.010.451.463.397.557.933.641.540.104.031	34
64	15.319.707.002.413.218	17
64	1.173.467.113.198.943.003.498.913.117.643.711	34
63	6.129.997.751.106.572	16
63	187.884.362.142.858.181.858.241.263.488.781	33
62	4.896.992.615.173.413	16
62	119.902.683.365.314.737.413.563.386.209.911	33
61	568.308.757.762.911	15
61	1.614.874.220.750.118.110.570.224.784.161	31
60	486.114.270.347.446	15
60	1.181.535.419.177.151.517.719.145.351.811	31
59	447.341.903.249.999	15
59	1.000.573.892.016.659.566.102.983.750.001	31
58	173.241.474.816.089	15
58	150.063.042.982.268.862.289.240.360.051	30
57	6.120.951.981.227	13
57	187.330.265.782.464.287.562.033.781	27
56	5.776.382.076.836	13
56	166.832.949.487.989.784.949.238.661	27
55	5.587.098.338.769	13
55	156.078.339.235.404.532.933.870.651	27
54	Prime! 5.537.131.346.429	13
54	153.299.117.738.060.837.711.992.351	27
53	4.517.859.004.184	13
53	102.055.249.908.454.809.942.550.201	27
52	4.489.128.926.599	13
52	100.761.392.598.161.895.293.167.001	27
51	1.828.395.133.423	13
51	16.715.143.819.633.691.834.151.761	26
50	1.487.237.428.873	13
50	11.059.375.849.211.294.857.395.011	26
49	611.885.676.707	12
49	1.872.020.406.798.976.040.202.781	25
48	611.555.414.387	12
48	1.870.000.124.333.334.210.000.781	25
47	555.263.013.934	12
47	1.541.585.073.218.123.705.851.451	25
46	553.975.714.845	12
46	1.534.445.463.192.913.645.444.351	25
45	183.321.776.691	12
45	168.034.369.046.640.963.430.861	24
44	164.404.707.182	12
44	135.144.538.718.817.835.441.531	24
43	145.196.422.724	12
43	105.410.005.859.958.500.014.501	24
42	56.734.942.971	11
42	16.094.268.769.896.786.249.061	23
41	18.346.234.883	11
41	1.682.921.672.002.761.292.861	22
40	4.623.988.479	10
40	Prime! 106.906.347.292.743.609.601	21
39	1.824.006.163	10
39	16.634.992.422.429.943.661	20
38	1.615.528.242	10
38	13.049.657.511.575.694.031	20
37	571.550.191	9
37	1.633.348.107.018.433.361	19
36	560.721.949	9
36	1.572.045.523.255.402.751	19
35	459.439.100	9
35	1.055.421.435.341.245.501	19
34	455.665.564	9
34	1.038.155.533.355.518.301	19
33	141.782.375	9
33	100.511.210.012.115.001	18
32	57.461.696	8
32	16.509.232.823.290.561	17
31	54.783.910	8
31	15.006.384.248.360.051	17
30	45.862.300	8
30	10.516.753.035.761.501	17
29	17.753.525	8
29	1.575.938.338.395.751	16
28	5.569.114	7
28	155.075.181.570.551	15
27	5.535.554	7
27	153.211.818.112.351	15
26	4.639.740	7
26	Prime! 107.635.959.536.701	15
25	4.565.455	7
25	Prime! 104.216.919.612.401	15
24	1.545.586	7
24	11.944.188.144.911	14
23	557.085	6
23	1.551.721.271.551	13
22	555.534	6
22	1.543.092.903.451	13
21	Prime! 475.081	6
21	Prime! 1.128.512.158.211	13
20	459.075	6
20	1.053.751.573.501	13
19	449.620	6
19	1.010.792.970.101	13
18	60.352	5
18	18.212.121.281	11
17	58.168	5
17	16.917.871.961	11
16	55.665	5
16	15.493.239.451	11
15	55.445	5
15	15.371.017.351	11
14	51.982	5
14	13.510.901.531	11
13	48.026	5
13	11.532.723.511	11
12	4.664	4
12	108.787.801	9
11	1.912	4
11	18.288.281	8
10	1.769	4
10	15.655.651	8
9	612	3
9	1.875.781	7
8	565	3
8	Prime! 1.598.951	7
7	554	3
7	1.537.351	7
6	174	3
6	152.251	6
5	144	3
5	104.401	6
4	Prime! 5	1
4	Prime! 151	3
3	4	1
3	Prime! 101	3
2	1	1
2	Prime! 11	2
1	0	1
1	1	1

	[CP22] Case X = 22 Palindromic Centered Icosidigonals
One can find the regular numbers of the form 11n² – 11n + 1 at %N 11n² – 11n + 1. under A069173. The palindromic numbers of the form 11n² – 11n + 1 are categorised as follows : %N 11n² – 11n + 1 is a palindrome. under A196494. %N Palindromes of the form 11n² – 11n + 1. under A195902. Nevertheless the following table will use {Formula 1 = 11n² + 11n + 1} for its base numbers (m=n+1).

	Index 1 up to 14 by Patrick De Geest [ not dated ] Index 15 up to 21 by Robert Xiao [ November 11, 2022 ]
21	342.984.461.435.878.825.320	21
21	1.294.021.748.651.058.340.294.920.438.501.568.471.204.921	43
20	33.647.953.619.813.640.820	20
20	12.454.032.610.812.428.837.773.882.421.801.623.045.421	41
19	8.043.312.475.603.102.332	19
19	711.643.631.382.117.573.202.375.711.283.136.346.117	39
18	5.865.978.534.325.259.951	19
18	378.506.745.816.811.974.373.479.118.618.547.605.873	39
17	4.030.319.882.810.327.729	19
17	178.678.261.935.538.792.272.297.835.539.162.876.871	39
16	534.050.327.583.715.456	18
16	3.137.307.276.315.011.691.961.105.136.727.037.313	37
15	397.048.630.777.431.645	18
15	1.734.123.767.224.565.641.465.654.227.673.214.371	37
14	3.987.885.154.604.870	16
14	174.935.508.069.497.030.794.960.805.539.471	33
13	39.586.343.600.545	14
13	Prime! 17.237.864.596.264.946.269.546.873.271	29
12	38.973.713.724.439	14
12	16.708.453.976.220.202.267.935.480.761	29
11	593.651.537.078	12
11	3.876.643.622.232.322.263.466.783	25
10	4.192.653.609	10
10	193.361.787.181.787.163.391	21
9	565.390.536	9
9	Prime! 3.516.331.046.401.336.153	19
8	560.670.366	9
8	3.457.863.858.583.687.543	19
7	422.820.434	9
7	1.966.548.318.138.456.691	19
6	362.349.815	9
6	1.444.271.276.721.724.441	19
5	Prime! 358.542.859	9
5	1.414.082.803.082.804.141	19
4	321.895.110	9
4	1.139.781.083.801.879.311	19
3	524.961	6
3	3.031.430.341.303	13
2	4.155	4
2	189.949.981	9
1	0	1
1	1	1

	[CP37] Case X = 37 Palindromic Centered Triacontaheptagonals
One can find the regular numbers of the form (37m² + 37m + 2) / 2 at %N (37m² + 37m +2) / 2. under A??????. The palindromic numbers of the form (37m² + 37m + 2) / 2 are categorised as follows : %N (37m² + 37m +2) / 2 is a palindrome. under A??????. %N Palindromes of the form (37m² + 37m +2) / 2. under A196496.

	Index 1 up to 11 by Patrick De Geest [ not dated ] Index 12 up to 14 by Robert Xiao [ November 11, 2022 ]
14	65.319.969.482.232.975.972	20
14	78.933.920.643.457.175.330.303.357.175.434.602.933.987	41
13	4.402.497.983.205.085.656	19
13	358.566.787.104.309.663.343.366.903.401.787.665.853	39
12	255.727.928.488.121.895	18
12	1.209.840.308.063.280.743.470.823.608.030.489.021	37
11	652.430.125.248.507	15
11	7.874.803.764.137.988.897.314.673.084.787	31
10	651.249.568.712.652	15
10	7.846.331.013.845.693.965.483.101.336.487	31
9	2.945.902.781.160	13
9	160.549.349.126.909.621.943.945.061	27
8	581.098.637.574	12
8	6.246.999.091.932.391.909.996.426	25
7	6.792.833	7
7	853.637.858.736.358	15
6	3.443.282	7
6	219.339.595.933.912	15
5	2.835.584	7
5	148.749.979.947.841	15
4	407.156	6
4	Prime! 3.066.863.686.603	13
3	Prime! 2.399	4
3	106.515.601	9
2	377	3
2	2.636.362	7
1	0	1
1	1	1

	[CP56] Case X = 56 Palindromic Centered Pentacontahexagonals
One can find the regular numbers of the form (56m² + 56m + 2) / 2 at %N (56m² + 56m +2) / 2. under A??????. The palindromic numbers of the form (56m² + 56m + 2) / 2 are categorised as follows : %N (56m² + 56m +2) / 2 is a palindrome. under A??????. %N Palindromes of the form (56m² + 56m +2) / 2. under ??????.

	Index 1 up to 4 by Patrick De Geest [ not dated ] Index 5 up to 11 by Robert Xiao [ November 11, 2022 ] Index 12 by Robert Xiao [ December 6, 2022 ] Index 13 by Patrick De Geest [ July 24, 2023 ]
13	193.122.537.707.602.417.516.558.324	27
13	1.044.296.807.977.480.893.274.968.131.318.694.723.980.847.797.086.924.401	55
12	6.648.271.278.671.614.717.965	22
12	1.237.586.307.854.677.394.967.227.694.937.764.587.036.857.321	46
11	803.883.779.463.703.526.385	21
11	18.094.415.664.775.753.169.099.096.135.757.746.651.449.081	44
10	256.429.228.529.531.525.689	21
10	1.841.166.578.839.019.738.567.658.379.109.388.756.611.481	43
9	5.275.633.207.777.375.418	19
9	779.304.560.804.095.199.828.991.590.408.065.403.977	39
8	166.586.585.621.988.593	18
8	777.030.534.257.379.830.038.973.752.435.030.777	36
7	159.479.946.814.558.583	18
7	712.147.896.207.284.165.561.482.702.698.741.217	36
6	66.528.144.051.921.965	17
6	123.927.830.627.811.839.938.118.726.038.729.321	36
5	2273238558823830	16
5	144.693.179.269.056.484.650.962.971.396.441	33
4	16.819.209.213	11
4	7.920.802.359.889.532.080.297	22
3	8.073.041.785	10
3	1.824.872.102.772.012.784.281	22
2	Curio! 709.000.780	9
2	14.075.098.988.989.057.041	20
1	0	1
1	1	1

	[CP666] Case X = 666 Palindromic Centered 666-gonals
One can find the regular numbers of the form (666m² + 666m + 2) / 2 at %N (666m² + 666m +2) / 2. under A??????. The palindromic numbers of the form (666m² + 666m + 2) / 2 are categorised as follows : %N (666m² + 666m +2) / 2 is a palindrome. under A??????. %N Palindromes of the form (666m² + 666m +2) / 2. under ??????.

	Index 1 up to 5 by Patrick De Geest [ not dated ] Index 6 up to 10 by Robert Xiao [ November 11, 2022 ] Index 11 by Robert Xiao [ December 6, 2022 ] Index 12 and 13 by Patrick De Geest [ July 24, 2023 ]
13	72.656.305.325.617.563.395.666.934	26
13	1.757.886.588.288.597.913.928.910.304.030.198.293.197.958.828.856.887.571	55
12	67.145.969.632.707.916.994.334.350	26
12	1.501.357.552.226.205.742.556.723.857.583.276.552.475.026.222.557.531.051	55
11	702.031.747.223.930.375.942.779	24
11	164.118.575.178.724.730.211.329.414.923.112.037.427.871.575.811.461	51
10	6.419.149.057.186.399.585	19
10	13.721.423.047.919.555.217.771.255.591.974.032.412.731	41
9	1.478.708.655.372.283.766	19
9	728.130.902.728.478.190.979.091.874.827.209.031.827	39
8	1.471.862.408.197.328.658	19
8	721.404.189.905.258.431.080.134.852.509.981.404.127	39
7	723.852.613.033.346.490	18
7	174.479.547.596.602.817.969.718.206.695.745.974.471	39
6	15.201.525.515.907.826	17
6	76.951.763.877.595.304.640.359.577.836.715.967	35
5	68.948.602.292.174	14
5	1.583.051.949.428.802.088.249.491.503.851	31
4	154.374.971	9
4	7.935.933.397.933.395.397	19
3	67.225	5
3	1.504.916.194.051	13
2	63.814	5
2	1.356.072.706.531	13
1	0	1
1	1	1

Further Sources Revealed

From the OEIS Wiki pages

https://oeis.org/wiki/Centered_polygonal_numbers

From the On-line Encyclopedia of Integer Sequences

A196495 Smallest multidigit palindromic centered n-gonal number

About palindromic centered decagonal numbers which are prime as well !

%N 5*n² + 5*n + 1 is a palindromic prime. under A134463.

%N Palindromic primes of the form 5*n² + 5*n + 1. under A134462.

Information from Wikipedia

General index Category:Figurate numbers

https://en.wikipedia.org/wiki/Centered_triangular_number

https://en.wikipedia.org/wiki/Centered_decagonal_number

From Terry Trotter's (†) pages

Polygonal Numbers, from the Wayback machine

Polygonal Numbers reconstructed website (only local for now)

Polygonal Numbers from 'teherba.org', an archive copy of the original webpage of 2004

A book or e-Book

“Figurate Numbers” by Michel-marie Deza and Elena Deza

Editor : World Scientific, 476 pages

Link : https://play.google.com/store/books/details?id=ERS7CgAAQBAJ

Chapter 4.7 treats some palindromic figurate numbers (page 277 to page 282).

Contributions

[ November 11, 2022 ]
Robert Xiao (email) added

six new centered triangular entries from [31] up to [36].

five new centered pentagonal entries from [39] up to [43].

two new centered hexagonal entries [25] and [26].

two new centered heptagonal entries [36] and [37].

ten new centered decagonal entries from [87] up to [96].

seven new centered icosidigonals (22-gonals) entries from [15] up to [21].

three new centered triacontaheptagonals (37-gonals) entries from [12] up to [14].

seven new centered 56-gonals entries from [5] up to [11]. See Plain Text Centered.txt only and A196495.

five new centered 666-gonals entries from [6] up to [10]. See also Plain Text Centered.txt only.

[ December 6, 2022 ]
Robert Xiao (email) added

five new centered triangular entries from [37] up to [41].

seven new centered squares entries from [53] up to [59].

eleven new centered pentagonal entries from [44] up to [54].

four new centered hexagonal entries [27] up to [30].

two new centered heptagonal entries [38] up to [46].

Robert provided (CP9) entry [47] that should also be a palindromic triangular missed by Feng Yuan plus three new centered nonagonals entries [49] up to [51].

fourteen new centered decagonal entries from [97] up to [110].

one new centered 56-gonals entry [12]. See Plain Text Centered.txt only and A196495.

one new centered 666-gonals entry from [11]. See also Plain Text Centered.txt only.

[ July 24, 2023 ]
Patrick De Geest (email) added

six new centered triangular entries from [42] up to [47].

two new centered pentagonal entries from [55] and [56].

seven new centered hexagonal entries [31] up to [37].

five new centered heptagonal entries [47] up to [51].

provided four new centered nonagonals entries [52] up to [55].

five new centered decagonal entries from [111] up to [115].

no new centered icosidigonals (CP22) up to length 55.

no new centered triacontaheptagonals (CP37) up to length 55.

one new centered pentacontahexagonal (CP56) entry [13].

two new centered 666-gonals entry from [12] up to [13].

[ TOP OF PAGE]

Centered Triangulars {Formula 1} Centered Triangulars {Formula 2}	(3n² + 3n + 2) / 2 (3m² – 3m + 2) / 2	Gen. Form. is Sn² +/– Sn + 2) / 2	Centered Decagonals {Formula 1} Centered Decagonals {Formula 2}	(10n² + 10n + 2) / 2 (10m² – 10m + 2) / 2	5base² + 5base + 1
Centered Squares {Formula 1} Centered Squares {Formula 2}	(4n² + 4n + 2) / 2 (4m² – 4m + 2) / 2	2base² + 2base + 1 OR base² + (base+1)²	Centered 22-gonal numbers [ Because it is in the OEIS ]	(22n² + 22n + 2) / 2 (22m² – 22m + 2) / 2	11base² + 11base + 1
Centered Pentagonals {Formula 1} Centered Pentagonals {Formula 2}	(5n² + 5n + 2) / 2 (5m² – 5m + 2) / 2		Centered 37-gonal numbers [ Because it is in the OEIS ]	(37n² + 37n + 2) / 2 (37m² – 37m + 2) / 2
Centered Hexagonals {Formula 1} Centered Hexagonals {Formula 2}	(6n² + 6n + 2) / 2 (6m² – 6m + 2) / 2	3base² + 3base + 1	Centered 56-gonal numbers [ Because it should be in the OEIS ]	(56n² + 56n + 2) / 2 (56m² – 56m + 2) / 2	28base² + 28base + 1
Centered Heptagonals {Formula 1} Centered Heptagonals {Formula 2}	(7n² + 7n + 2) / 2 (7m² – 7m + 2) / 2		Centered 666-gonal numbers [ Because 666 is a fetish number ]	(666n² + 666n + 2) / 2 (666m² – 666m + 2) / 2	333base² + 333base + 1
Centered Octagonals {Formula 1} Centered Octagonals {Formula 2}	(8n² + 8n + 2) / 2 (8m² – 8m + 2) / 2	4base² + 4base + 1 OR (2*base + 1)²
Centered Nonagonals {Formula 1} Centered Nonagonals {Formula 2}	(9n² + 9n + 2) / 2 (9m² – 9m + 2) / 2		PLAIN TEXT CENTERED