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



Introduction

Palindromic Centered Polygonals are defined and calculated by these extraordinary intricate and excruciatingly complex formulae :)

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

Palindromic numbers are numbers which read the same from
 p_right left to right (forwards) as from the right to left (backwards) p_left
Here are a few random examples : 7, 3113, 44611644


Palindromic Centered Polygonals

To go from Formula 2 = m Formula 1 = n.
Proof by substitution whereby  m  equals  n + 1 
( S*m^2 S*m + 2 ) / 2 =
      ( S*(n+1)^2 – S*(n+1) + 2 ) / 2 =
      ( S*(n^2+2n+1) – S*(n+1) + 2 ) / 2 =
      ( S*n^2 + 2*S*n + S – S*n – S + 2 ) / 2 =  
      ( S*n^2 + S*n + S – S + 2 ) / 2 =
( S*n^2 + S*n + 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

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

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

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 ?

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 5, 8, 9, 12, 13, 19, 24, 25, 30, 32, 33, 40, 43, 44, 45, 47

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

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

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


The Table


Index Nr BasenumberLength
Palindromic Centered PolygonalsLength
halt down
Case X = 3 Palindromic Centered Triangulars
One can find the regular numbers of the form {Formula 2} (3*m2 – 3*m + 2) / 2 at
%N (3*m2 – 3*m + 2) / 2. under A005448.
The palindromic numbers of the form (3*m2 – 3*m + 2) / 2 are categorised as follows :
%N (3*m2 – 3*m + 2) / 2 is a palindrome. under A195903.
%N Palindromes of the form (3*m2 – 3*m + 2) / 2. under A162703.

Nevertheless the following table will use {Formula 1 = (3*n2 + 3*n + 2) / 2} for
its base numbers (m=n+1).
   
Index 1 to 17 were taken from the OEIS (See above)
Index 17 to 30 by Patrick De Geest [ Last one May 15, 2021 ]
3017.729.422.815.023.433.06120
471.498.650.030.810.150.303.051.018.030.056.894.17439
298.781.786.523.212.541.11519
115.679.661.808.916.116.404.611.619.808.166.976.51139
281.698.666.724.874.603.13319
4.328.202.963.294.315.983.895.134.923.692.028.23437
271.144.251.252.690.582.76419
1.963.966.393.925.951.835.381.595.293.936.693.69137
2682.738.939.025.750.80017
Prime!    10.268.598.046.660.363.236.306.664.089.586.20135
2517.108.271.682.888.34117
439.039.439.963.278.626.872.369.934.930.93433
2410.020.895.012.660.70017
150.627.505.282.152.151.251.282.505.726.05133
23213.031.868.452.70915
68.073.865.464.678.787.646.456.837.08629
22189.344.689.666.18715
53.777.117.257.177.277.175.271.177.73529
21181.002.188.703.85315
49.142.688.473.378.087.337.488.624.19429
20170.438.376.135.14115
43.573.860.089.375.957.398.006.837.53429
1982.406.037.714.59914
10.186.132.577.729.992.777.523.168.10129
1817.476.134.422.42614
458.122.911.526.080.625.119.221.85427
172.036.493.584.52513
6.220.959.179.720.279.719.590.22625
16Prime!    196.759.830.16712
58.071.646.151.315.164.617.08523
15167.526.192.48612
42.097.537.753.535.773.579.02423
141.702.002.46610
4.345.218.593.958.125.43419
13215.198.6949
69.465.717.171.756.49617
1217.880.5868
479.573.060.375.97415
11Prime!    9.531.9797
136.287.949.782.63115
1020.7945
648.616.8469
9Prime!    2.0694
6.424.2467
81.8214
4.976.7947
72563
Prime!    98.6895
62483
92.6295
52103
66.4665
4Prime!    1733
45.1545
31003
15.1515
211
41
101
11
up down
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 to 52 by Patrick De Geest [ Last one June 16, 2021 ]
5289.940.695.078.362.814.44420
16.178.657.262.358.073.985.458.937.085.326.275.687.16141
5184.753.839.053.503.067.85420
14.366.426.468.614.203.601.510.630.241.686.462.466.34141
5084.284.587.587.769.426.72020
14.207.783.409.680.752.220.402.225.708.690.438.770.24141
49124.566.258.570.164.69718
31.033.505.548.338.259.895.283.384.550.533.01335
4891.424.589.460.359.85517
16.716.911.115.990.684.748.609.951.111.961.76135
4789.663.751.709.213.50517
16.079.176.741.142.975.657.924.114.767.197.06135
4672.421.835.667.809.20017
10.489.844.562.990.321.812.309.926.544.898.40135
1 - 45See list at Sumsquare.htm 
up down
Case X = 5 Palindromic Centered Pentagonals
One can find the regular numbers of the form (5*n2 + 5*n + 2) / 2 at
%N (5*n2 + 5*n + 2) / 2. under A005891.
The palindromic numbers of the form (5*n2 + 5*n + 2) / 2 are categorised as follows :
%N (5*n2 + 5*n + 2) / 2 is a palindrome. under A??????.
%N Palindromes of the form (5*n2 + 5*n + 2) / 2. under A??????.
388.545.229.924.052.920.92819
182.552.386.137.323.721.949.127.323.731.683.255.28139
377.411.900.058.658.854.46819
Prime!    137.340.656.198.867.825.777.528.768.891.656.043.73139
366.548.959.543.466.373.94419
107.222.177.754.898.242.595.242.898.457.771.222.70139
356.455.494.583.580.369.03919
104.183.525.796.588.705.676.507.885.697.525.381.40139
346.334.531.182.684.034.79919
100.315.713.260.990.991.646.199.099.062.317.513.00139
33766.048.834.768.223.79218
1.467.077.043.124.383.593.953.834.213.407.707.64137
32644.147.472.408.716.10418
1.037.314.915.526.344.187.814.436.255.194.137.30137
31158.641.107.117.468.12918
62.917.502.168.639.993.039.993.686.120.571.92635
30156.906.204.779.292.75718
61.548.892.745.603.383.438.330.654.729.884.51635
2976.607.615.255.405.80717
14.671.816.787.800.711.511.700.878.761.817.64135
2874.745.660.814.331.48317
13.967.284.526.427.722.622.772.462.548.276.93135
2765.323.181.999.789.04017
10.667.795.266.443.907.270.934.466.259.776.60135
267.963.013.978.107.11516
158.523.979.038.823.272.328.830.979.325.85133
251.615.286.421.727.42616
6.522.875.560.542.483.842.450.655.782.25631
24647.673.577.893.76015
1.048.702.658.754.262.624.578.562.078.40131
2379.049.055.962.16414
15.621.883.121.273.537.212.138.812.65129
2215.846.566.505.00914
627.784.174.994.222.499.471.487.72627
217.512.476.533.59213
141.093.259.169.444.961.952.390.14127
207.435.245.752.98813
138.207.198.518.333.815.891.702.83127
191.553.657.288.04113
6.034.627.421.711.171.247.264.30625
18793.717.097.72412
Prime!    1.574.967.078.050.508.707.694.75125
1765.751.464.70411
10.808.137.776.967.773.180.80123
161.559.881.23310
6.083.073.656.563.703.80619
15778.794.8609
1.516.303.586.853.036.15119
1476.591.4878
14.665.639.893.656.64117
1315.759.1108
620.873.909.378.02615
1215.534.4418
603.297.181.792.30615
11Prime!    8.528.9837
181.858.898.858.18115
107.834.1407
153.434.393.434.35115
9780.6996
1.523.729.273.25113
8737.0036
1.357.935.397.53113
78.7874
193.050.3919
66.4644
104.474.4019
5642
104015
481
Prime!    1813
3Prime!    71
1413
211
61
101
11
up down
Case X = 6 Palindromic Centered Hexagonals
One can find the regular numbers of the form 3*n2 + 3*n + 1 at
%N 3*n2 + 3*n + 1. under A003215.
The palindromic numbers of the form 3*n2 + 3*n + 1 are categorised as follows :
%N 3*n2 + 3*n + 1 is a palindrome. under A??????.
%N Palindromes of the form 3*n2 + 3*n + 1. under A??????.
2417.422.294.995.888.404.72220
910.609.088.771.274.444.979.444.472.177.880.906.01939
2315.426.989.438.535.955.89120
713.976.009.410.099.782.797.287.990.014.900.679.31739
227.781.806.191.361.855.16419
181.669.522.799.753.105.999.501.357.997.225.966.18139
21731.038.731.693.350.99518
1.603.252.881.707.469.675.769.647.071.882.523.06137
20179.258.981.327.928.41218
96.401.347.160.179.761.716.797.106.174.310.46935
19153.205.888.386.457.27318
70.416.132.708.850.811.011.805.880.723.161.40735
1816.096.354.798.508.23117
777.277.913.398.376.909.673.893.319.772.77733
177.060.467.049.615.58016
149.550.584.876.122.020.221.678.485.055.94133
16671.640.175.743.58915
1.353.301.577.018.639.368.107.751.033.53131
15160.017.597.111.65815
76.816.894.156.167.176.165.149.861.86729
1479.625.800.994.00514
19.020.804.551.810.901.815.540.802.09129
13583.937.096.77512
1.022.947.598.971.798.957.492.20125
1272.356.279.62511
15.706.293.603.730.639.260.75123
1116.208.835.47811
788.179.042.707.240.971.88721
106.617.5147
131.374.494.473.13115
91.750.6177
9.193.984.893.91913
87.7604
180.676.0819
77.4794
167.828.7619
68053
1.946.4917
56293
1.188.8117
46003
1.081.8017
3Prime!    172
Prime!    9193
211
Prime!    71
101
11
up down
Case X = 7 Palindromic Centered Heptagonals
One can find the regular numbers of the form (7*m2 – 7*m + 2) / 2) at
%N (7*m2 – 7*m + 2) / 2). under A069099.
The palindromic numbers of the form (7*m2 – 7*m + 2) / 2) are categorised as follows :
%N (7*m2 – 7*m + 2) / 2) is a palindrome. under A??????.
%N Palindromes of the form (7*m2 – 7*m + 2) / 2). under A??????.

Nevertheless the following table will use {Formula 1 = (7*n2 + 7*n + 2) / 2} for
its base numbers (m=n+1).
35932.758.758.091.321.91618
3.045.136.152.786.228.195.918.226.872.516.315.40337
34566.388.600.619.580.46018
Prime!    1.122.786.164.191.323.168.613.231.914.616.872.21137
3343.142.652.872.006.56917
6.514.509.738.920.598.448.950.298.379.054.15634
3222.442.486.416.824.77917
1.762.828.187.992.776.556.772.997.818.282.67134
318.028.245.530.124.52216
225.584.542.021.875.313.578.120.245.485.52233
302.266.182.015.260.97916
17.974.533.242.023.100.132.024.233.547.97132
291.035.206.065.266.85616
3.750.780.591.478.505.058.741.950.870.57331
28733.722.882.837.28015
1.884.222.440.796.673.766.970.442.224.88131
27709.830.923.358.77915
1.763.509.789.147.321.237.419.879.053.67131
26137.171.486.712.36915
65.856.058.684.086.168.048.685.065.85629
25136.845.415.594.29415
65.543.337.192.113.131.129.173.334.55629
2470.267.104.137.19514
17.281.130.733.396.169.333.703.118.27129
2330.771.414.144.24814
3.314.079.749.528.998.259.479.704.13328
22Prime!    20.397.301.735.15914
1.456.174.713.262.992.623.174.716.54128
2114.056.310.919.25414
691.529.568.305.636.503.865.925.19627
205.525.124.851.02413
106.844.516.167.929.761.615.448.60127
192.308.275.787.91913
18.648.479.895.833.859.897.484.68126
181.875.549.816.87913
12.311.904.904.588.540.940.911.32126
17888.606.676.06212
2.763.676.386.599.956.836.763.67225
16639.152.410.41512
1.429.805.313.089.803.135.089.24125
15155.707.617.28112
84.857.017.278.187.271.075.84823
1466.854.165.72411
15.643.178.161.516.187.134.65123
133.369.744.49610
39.743.122.900.922.134.79320
12886.441.7829
2.750.226.618.166.220.57219
11648.289.9599
1.470.979.550.559.790.74119
1094.660.3368
31.362.027.572.026.31317
93.313.9237
38.437.311.373.48314
81.390.8497
6.770.618.160.77613
7Prime!    798.7576
2.233.047.403.32213
632.4365
3.682.442.86310
57403
1.919.1917
4Prime!    192
1.3314
3Prime!    21
222
211
81
101
11
up down
Case X = 8 Palindromic Centered Octagonals
One can find the regular numbers of the form 4*n2 + 4*n + 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??????.
allThis is a subset of the palindromic squares
namely every palindromic square whose base is “odd”
Baseocta becomes ( base bsquare – 1 divided by two ).
Let me list them up to length 13.
!
381.534.6537
9.420.645.460.24913
371.147.3377
5.265.533.355.62513
36635.4346
1.615.108.015.16113
35555.5556
1.234.567.654.32113
34555.0556
1.232.346.432.32113
33554.5556
1.230.127.210.32113
32551.0056
1.214.428.244.12113
31550.5056
1.212.225.222.12113
30550.0056
1.210.024.200.12113
29521.0756
1.086.078.706.80113
28506.0506
1.024.348.434.20113
27505.5506
1.022.325.232.20113
26505.0506
1.020.304.030.20113
25501.0006
1.004.006.004.00113
24500.5006
1.002.003.002.00113
23500.0006
1.000.002.000.00113
2255.5555
12.345.654.32111
2155.0055
12.102.420.12111
2050.5505
10.221.412.20111
1950.0005
10.000.200.00111
1815.3465
942.060.2499
1711.4325
522.808.2259
165.6054
125.686.5219
155.5554
123.454.3219
145.5054
121.242.1219
135.1004
104.060.4019
125.0504
102.030.2019
115.0004
100.020.0019
101.1424
5.221.2257
95553
1.234.3217
85003
1.002.0017
71533
94.2495
6602
14.6415
5552
12.3215
4502
10.2015
3Prime!    51
1213
211
91
101
11
up down
Case X = 9 Palindromic Centered Nonagonals
One can find the regular numbers of the form (9*m2 – 9*m + 2) / 2 at
%N (9*m2 – 9*m +2) / 2. under A060544.
The palindromic numbers of the form (9*m2 – 9*m + 2) / 2 are categorised as follows :
%N (9*m2 – 9*m +2) / 2 is a palindrome. under A??????.
%N Palindromes of the form (9*m2 – 9*m +2) / 2. under A??????.

Nevertheless the following table will use {Formula 1 = (9*n2 + 9*n + 2) / 2} for
its base numbers (m=n+1).
allThis is a subset of the palindromic triangulars
namely every palindromic triangular with
base bnona equal to (btria – 1) divisible by 3.
Let me list them up to length 47.
!
4738.286.913.835.022.303.780.76523
6.596.494.969.546.900.318.903.773.098.130.096.459.694.946.95646
46258.593.922.027.612.151.86721
300.918.674.293.302.389.819.918.983.203.392.476.819.00342
45111.156.781.896.700.878.02821
55.601.235.727.338.276.212.021.267.283.372.753.210.65541
4411.789.941.359.940.700.95020
625.512.227.718.781.732.353.237.187.817.722.215.52639
433.888.611.090.993.192.10519
68.045.832.976.478.686.977.968.687.467.923.854.08638
421.217.414.068.855.875.34519
6.669.436.567.716.980.985.890.896.177.656.349.66637
41Prime!    1.212.527.593.913.419.52919
6.616.004.247.006.598.875.788.956.007.424.006.16637
401.079.177.623.768.796.59119
5.240.809.546.394.698.286.828.964.936.459.080.42537
39258.509.259.706.890.10718
300.721.668.093.919.607.706.919.390.866.127.00336
38117.483.227.172.259.62518
62.110.389.000.639.430.803.493.600.098.301.12635
3737.513.595.580.544.39417
6.332.714.340.212.879.669.782.120.434.172.33634
3636.033.401.143.279.97617
5.842.826.990.786.388.228.836.870.996.282.48534
3526.119.645.131.833.61217
3.070.061.378.158.136.996.318.518.731.600.70334
341.354.087.421.826.19716
8.250.987.356.765.633.365.676.537.890.52831
33120.928.470.687.90915
65.806.627.603.124.642.130.672.660.85629
3236.412.576.729.76314
5.966.440.848.454.114.548.480.446.69528
3113.933.969.125.85714
873.699.730.201.575.102.037.996.37827
303.621.907.975.01113
59.031.978.207.533.570.287.913.09526
291.684.040.145.43913
12.761.960.451.533.515.406.916.72126
28135.090.111.24212
82.122.021.699.799.612.022.12823
27Prime!    88.814.814.88711
35.496.321.045.754.012.369.45323
26Prime!    82.214.815.54711
30.416.741.529.792.514.761.40323
2520.915.965.57911
1.968.649.272.552.729.468.69122
248.898.308.55210
356.309.527.929.725.903.65321
23Prime!    8.296.010.22710
309.707.035.626.530.707.90321
225.842.580.30010
153.610.850.555.058.016.35121
211.837.866.92410
15.199.896.744.769.899.15120
20338.236.2689
514.816.979.979.618.41518
19122.835.2509
67.898.244.444.289.87617
1849.907.0198
11.208.197.679.180.21117
1719.669.9398
1.741.079.339.701.47116
1612.326.3698
683.727.232.727.38615
1512.309.9698
681.909.070.909.18615
1411.890.1308
636.188.414.881.63615
1311.568.3908
602.224.464.422.20615
1210.950.3238
539.593.131.395.93515
114.270.7977
82.078.700.787.02814
1059.7195
16.048.884.06111
955.6845
13.953.435.93111
819.0555
1.634.004.36110
716.7605
1.264.114.62110
6Prime!    8873
3.544.4537
53703
617.7166
4362
5.9954
3Prime!    112
5953
2Prime!    31
552
101
11
up down
Case X = 10 Palindromic Centered Decagonals
One can find the regular numbers of the form 5*n2 + 5*n + 1 at
%N 5*n2 + 5*n + 1. under A062786.
The palindromic numbers of the form 5*n2 + 5*n + 1 are categorised as follows :
%N 5*n2 + 5*n + 1 is a palindrome. under A??????.
%N Palindromes of the form 5*n2 + 5*n + 1. under A??????.
The palindromic primes of the form 5*n2 + 5*n + 1 are categorised as follows :
%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.
866.036.281.167.110.772.12719
182.183.451.642.080.926.515.629.080.246.154.381.28139
856.033.764.984.134.155.59219
182.031.599.418.817.234.444.432.718.814.995.130.28139
845.760.242.508.141.202.32819
Prime!    165.901.968.762.984.246.868.642.489.267.869.109.56139
834.611.483.393.881.208.30419
106.328.895.460.210.736.868.637.012.064.598.823.60139
82612.734.698.031.971.33218
1.877.219.050.861.655.467.645.561.680.509.127.78137
81580.453.550.554.201.91618
1.684.631.621.754.897.200.027.984.571.261.364.86137
80560.645.404.450.209.85018
1.571.616.347.655.696.916.196.965.567.436.161.75137
79559.469.308.539.469.89418
1.565.029.535.988.162.807.082.618.895.359.205.65137
78473.572.348.147.542.59318
1.121.353.844.649.886.444.446.889.464.483.531.21137
77460.593.464.512.444.27918
1.060.731.697.757.881.339.331.887.577.961.370.60137
76459.728.007.205.710.32418
1.056.749.203.046.668.220.228.666.403.029.476.50137
75152.957.448.255.024.43318
116.979.904.883.442.385.583.244.388.409.979.61136
74144.078.685.966.944.23518
103.793.338.749.806.668.866.608.947.833.397.30136
73142.171.056.703.991.99518
101.063.046.821.648.536.635.846.128.640.360.10136
7256.473.953.204.117.76517
15.946.536.952.504.416.161.440.525.963.564.95135
7152.002.994.083.007.62217
13.521.556.967.986.628.982.668.976.965.512.53135
7045.723.888.363.764.68017
10.453.369.835.510.075.757.001.553.896.335.40135
6919.081.039.958.424.16717
1.820.430.429.474.898.778.984.749.240.340.28134
6818.126.473.120.621.46817
1.642.845.138.963.062.992.603.698.315.482.46134
6717.423.506.104.996.59417
1.517.892.824.954.267.997.624.594.282.987.15134
6617.404.751.100.767.93017
1.514.626.804.398.412.442.148.934.086.264.15134
6516.149.368.108.154.55717
1.304.010.451.463.397.557.933.641.540.104.03134
6415.319.707.002.413.21817
1.173.467.113.198.943.003.498.913.117.643.71134
636.129.997.751.106.57216
187.884.362.142.858.181.858.241.263.488.78133
624.896.992.615.173.41316
119.902.683.365.314.737.413.563.386.209.91133
61568.308.757.762.91115
1.614.874.220.750.118.110.570.224.784.16131
60486.114.270.347.44615
1.181.535.419.177.151.517.719.145.351.81131
59447.341.903.249.99915
1.000.573.892.016.659.566.102.983.750.00131
58173.241.474.816.08915
150.063.042.982.268.862.289.240.360.05130
576.120.951.981.22713
187.330.265.782.464.287.562.033.78127
565.776.382.076.83613
166.832.949.487.989.784.949.238.66127
555.587.098.338.76913
156.078.339.235.404.532.933.870.65127
54Prime!    5.537.131.346.42913
153.299.117.738.060.837.711.992.35127
534.517.859.004.18413
102.055.249.908.454.809.942.550.20127
524.489.128.926.59913
100.761.392.598.161.895.293.167.00127
511.828.395.133.42313
16.715.143.819.633.691.834.151.76126
501.487.237.428.87313
11.059.375.849.211.294.857.395.01126
49611.885.676.70712
1.872.020.406.798.976.040.202.78125
48611.555.414.38712
1.870.000.124.333.334.210.000.78125
47555.263.013.93412
1.541.585.073.218.123.705.851.45125
46553.975.714.84512
1.534.445.463.192.913.645.444.35125
45183.321.776.69112
168.034.369.046.640.963.430.86124
44164.404.707.18212
135.144.538.718.817.835.441.53124
43145.196.422.72412
105.410.005.859.958.500.014.50124
4256.734.942.97111
16.094.268.769.896.786.249.06123
4118.346.234.88311
1.682.921.672.002.761.292.86122
404.623.988.47910
Prime!    106.906.347.292.743.609.60121
391.824.006.16310
16.634.992.422.429.943.66120
381.615.528.24210
13.049.657.511.575.694.03120
37571.550.1919
1.633.348.107.018.433.36119
36560.721.9499
1.572.045.523.255.402.75119
35459.439.1009
1.055.421.435.341.245.50119
34455.665.5649
1.038.155.533.355.518.30119
33141.782.3759
100.511.210.012.115.00118
3257.461.6968
16.509.232.823.290.56117
3154.783.9108
15.006.384.248.360.05117
3045.862.3008
10.516.753.035.761.50117
2917.753.5258
1.575.938.338.395.75116
285.569.1147
155.075.181.570.55115
275.535.5547
153.211.818.112.35115
264.639.7407
Prime!    107.635.959.536.70115
254.565.4557
Prime!    104.216.919.612.40115
241.545.5867
11.944.188.144.91114
23557.0856
1.551.721.271.55113
22555.5346
1.543.092.903.45113
21Prime!    475.0816
Prime!    1.128.512.158.21113
20459.0756
1.053.751.573.50113
19449.6206
1.010.792.970.10113
1860.3525
18.212.121.28111
1758.1685
16.917.871.96111
1655.6655
15.493.239.45111
1555.4455
15.371.017.35111
1451.9825
13.510.901.53111
1348.0265
11.532.723.51111
124.6644
108.787.8019
111.9124
18.288.2818
101.7694
15.655.6518
96123
1.875.7817
85653
Prime!    1.598.9517
75543
1.537.3517
61743
152.2516
51443
104.4016
4Prime!    51
Prime!    1513
341
Prime!    1013
211
Prime!    112
101
11
up down
Case X = 22 Palindromic Centered Icosidigonals
One can find the regular numbers of the form 11*n2 – 11*n + 1 at
%N 11*n2 – 11*n + 1. under A069173.
The palindromic numbers of the form 11*n2 – 11*n + 1 are categorised as follows :
%N 11*n2 – 11*n + 1 is a palindrome. under A196494.
%N Palindromes of the form 11*n2 – 11*n + 1. under A195902.

Nevertheless the following table will use {Formula 1 = 11*n2 + 11*n + 1} for
its base numbers (m=n+1).
143.987.885.154.604.87016
174.935.508.069.497.030.794.960.805.539.47133
1339.586.343.600.54514
Prime!    17.237.864.596.264.946.269.546.873.27129
1238.973.713.724.43914
16.708.453.976.220.202.267.935.480.76129
11593.651.537.07812
3.876.643.622.232.322.263.466.78325
104.192.653.60910
193.361.787.181.787.163.39121
9565.390.5369
Prime!    3.516.331.046.401.336.15319
8560.670.3669
3.457.863.858.583.687.54319
7422.820.4349
1.966.548.318.138.456.69119
6362.349.8159
1.444.271.276.721.724.44119
5Prime!    358.542.8599
1.414.082.803.082.804.14119
4321.895.1109
1.139.781.083.801.879.31119
3524.9616
3.031.430.341.30313
24.1554
189.949.9819
101
11
up halt
Case X = 37 Palindromic Centered Triacontaheptagonals
One can find the regular numbers of the form (37*m2 + 37*m + 2) / 2 at
%N (37*m2 + 37*m +2) / 2. under A??????.
The palindromic numbers of the form (37*m2 + 37*m + 2) / 2 are categorised as follows :
%N (37*m2 + 37*m +2) / 2 is a palindrome. under A??????.
%N Palindromes of the form (37*m2 + 37*m +2) / 2. under A196496.
11652.430.125.248.50715
7.874.803.764.137.988.897.314.673.084.78731
10651.249.568.712.65215
7.846.331.013.845.693.965.483.101.336.48731
92.945.902.781.16013
160.549.349.126.909.621.943.945.06127
8581.098.637.57412
6.246.999.091.932.391.909.996.42625
76.792.8337
853.637.858.736.35815
63.443.2827
219.339.595.933.91215
52.835.5847
148.749.979.947.84115
4407.1566
Prime!    3.066.863.686.60313
3Prime!    2.3994
106.515.6019
23773
2.636.3627
101
11




Further Sources Revealed

From the OEIS Wiki pages
http://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

CP5{29} = 32218588608327889 * 455383597530039769 (2 distinct prime factors) - a nice semiprime!







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( © All rights reserved ) - Last modified : November 12, 2021.
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E-mail address : pdg@worldofnumbers.com