Palindromic Incremented Squares of the form n^2+X

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Palindromic Incremented Squares of the form n^2+x
n(n+0) n(n+1) n(n+2) n(n+x) n^2+1 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

Incremented Square numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base² + X

Palindromic Incremented Squares

Incremented Squares of the form x² + 0 can only start or end with a 0, 1, 4, 5, 6 or 9

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

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

4 can only be followed by an even number : 40, 42, 44, 46 or 48

5 can only be followed by 2 : 52

6 can only be followed by an odd number : 61, 63, 65, 67 or 69

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

Incremented Squares of the form x² + 1 can only start or end with a 0, 1, 2, 5, 6 or 7.

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 only be followed by 0 : 10

2 can only be followed by an even number : 20, 22, 24, 26 or 28

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

6 can only be followed by 2 : 62

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

Incremented Squares of the form x² + 2 can only start or end with a 1, 2, 3, 6, 7 or 8.

1 can only be followed by an odd number : 11, 13, 15, 17 or 19

2 can only be followed by 0 : 20

3 can only be followed by an even number : 30, 32, 34, 36 or 38

6 can only be followed by an even number : 60, 62, 64, 66 or 68

7 can only be followed by 2 : 72

8 can only be followed by an odd number : 81, 83, 85, 87 or 89

Incremented Squares of the form x² + 3 can only start or end with a 2, 3, 4, 7, 8 or 9.

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

2 can only be followed by an odd number : 21, 23, 25, 27 or 29

3 can only be followed by 0 : 30

4 can only be followed by an even number : 40, 42, 44, 46 or 48

7 can only be followed by an even number : 70, 72, 74, 76 or 78

8 can only be followed by 2 : 82

9 can only be followed by an odd number : 91, 93, 95, 97 or 99

Incremented Squares of the form x² + 4 can only start or end with a 0, 3, 4, 5, 8 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.

3 can only be followed by an odd number : 31, 33, 35, 37 or 39

4 can only be followed by 0 : 40

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

8 can only be followed by an even number : 80, 82, 84, 86 or 88

9 can only be followed by 2 : 92

Incremented Squares of the form x² + 5 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 only be followed by an even number : 10, 12, 14, 16 or 18

4 can only be followed by an odd number : 41, 43, 45, 47 or 49

5 can only be followed by 0 : 50

6 can only be followed by an even number : 60, 60, 64, 66 or 68

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

Incremented Squares of the form x² + 6 can only start or end with a 0, 1, 2, 5, 6 or 7.

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

1 can only be followed by 3 : 13

2 can only be followed by an even number : 20, 22, 24, 26 or 28

5 can only be followed by an odd number : 51, 53, 55, 57 or 59

6 can only be followed by 0 : 60

7 can only be followed by an even number : 70, 72, 74, 76 or 78

Incremented Squares of the form x² + 7 can only start or end with a 1, 2, 3, 6, 7 or 8.

1 can only be followed by an odd number : 11, 13, 15, 17 or 19

2 can only be followed by 3 : 23

3 can only be followed by an even number : 30, 32, 34, 36 or 38

6 can only be followed by an odd number : 61, 63, 65, 67 or 69

7 can only be followed by 0 : 70

8 can only be followed by an even number : 80, 82, 84, 86 or 88

Incremented Squares of the form x² + 8 can only start or end with a 2, 3, 4, 7, 8 or 9.

2 can only be followed by an odd number : 21, 23, 25, 27 or 29

3 can only be followed by 3 : 33

4 can only be followed by an even number : 40, 42, 44, 46 or 48

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

8 can only be followed by 0 : 80

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

Incremented Squares of the form x² + 9 can only start or end with a 0, 3, 4, 5, 8 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.

3 can only be followed by an odd number : 31, 33, 35, 37 or 39

4 can only be followed by 3 : 43

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

8 can only be followed by an odd number : 81, 83, 85, 87 or 89

9 can only be followed by 0 : 90

An infinite palindromic pattern hides in the list for case n^2 + 3

21² + 3 = 444
201² + 3 = 40404
2001² + 3 = 4004004
20001² + 3 = 400040004
200001² + 3 = 40000400004
2000001² + 3 = 4000004000004
20000001² + 3 = 400000040000004

Two infinite palindromic pattern hide in the list for case n^2 + 4

20² + 4 = 404
200² + 4 = 40004
2000² + 4 = 4000004
20000² + 4 = 400000004
200000² + 4 = 40000000004
2000000² + 4 = 4000000000004
20000000² + 4 = 400000000000004

2912² + 4 = 8479748
29012² + 4 = 841696148
290012² + 4 = 84106960148
2900012² + 4 = 8410069600148
29000012² + 4 = 841000696000148
290000012² + 4 = 84100006960000148
2900000012² + 4 = 8410000069600000148

A finite palindromic pattern resides in the list for case n^2 + 5

(SQ5) 30990992 [960441585144069] {L=15}
(SQ5) 30990990992 [960441522666225144069] {L=21}
(SQ5) 30990990990992 [960441522603747306225144069] {L=27}
(SQ5) 30990990990990992 [is no longer palindromic] {L=33}

An infinite palindromic pattern hides in the list for case n^2 + 8

31² + 8 = 969
301² + 8 = 90609
3001² + 8 = 9006009
30001² + 8 = 900060009
300001² + 8 = 90000600009
3000001² + 8 = 9000006000009
30000001² + 8 = 900000060000009

An infinite palindromic pattern hides in the list for case n^2 + 9

30² + 9 = 909
300² + 9 = 90009
3000² + 9 = 9000009
30000² + 9 = 900000009
300000² + 9 = 90000000009
3000000² + 9 = 9000000000009
30000000² + 9 = 900000000000009

The Table

Index Nr	Basenumber	Length
Index Nr	Palindromic Incremented Squares of form n^2+X	Length

	Case X = 0
See in-depth webpage about Palindromic Square Numbers

	Case X = 1
See in-depth webpage about Palindromic Incremented Squares

	Case X = 2
One can find the regular numbers of the form n² + 2 at %N n^2 + 2. under A059100. The palindromic numbers of the form n² + 2 are categorised as follows : %N n^2 + 2 is a palindrome. under A0?????. %N Palindromes of the form n^2 + 2. under A0?????.
68	817.683.802.207.308	15
68	668.606.800.392.199.991.293.008.606.866	30
67	791.470.648.804.932	15
67	626.425.787.919.700.007.919.787.524.626	30
66	364.768.101.292.427	15
66	133.055.767.720.482.284.027.767.550.331	30
65	284.676.297.563.004	15
65	81.040.594.394.179.997.149.349.504.018	29
64	186.797.158.329.471	15
64	34.893.178.359.965.456.995.387.139.843	29
63	141.426.468.677.100	15
63	20.001.446.042.474.747.424.064.410.002	29
62	140.737.100.319.083	15
62	19.806.931.406.223.632.260.413.960.891	29
61	133.805.135.860.437	15
61	17.903.814.382.630.003.628.341.830.971	29
60	93.455.496.826.376	14
60	8.733.929.887.064.774.607.889.293.378	28
59	55.349.225.724.449	14
59	3.063.536.788.296.006.928.876.353.603	28
58	45.492.981.517.640	14
58	2.069.611.367.364.334.637.631.169.602	28
57	14.363.881.876.060	14
57	206.321.102.549.404.945.201.123.602	27
56	13.413.161.997.313	14
56	179.912.914.766.161.667.419.219.971	27
55	10.902.121.823.947	14
55	118.856.260.264.181.462.062.658.811	27
54	9.114.951.055.994	13
54	83.082.332.753.166.135.723.328.038	26
54	2.925.054.461.666	13
54	8.555.943.603.712.173.063.495.558	25
53	2.688.021.576.515	13
53	7.225.459.995.810.185.999.545.227	25
52	1.954.611.231.091	13
52	3.820.505.064.707.074.605.050.283	25
51	1.861.509.029.379	13
51	3.465.215.866.459.546.685.125.643	25
50	268.803.849.765	12
50	72.255.509.648.484.690.555.227	23
49	250.793.696.732	12
49	62.897.478.320.502.387.479.826	23
48	144.598.910.530	12
48	20.908.844.926.462.944.880.902	23
47	142.019.320.310	12
47	20.169.487.341.314.378.496.102	23
46	11.645.196.023	11
46	135.610.590.414.095.016.531	21
45	8.487.412.695	10
45	72.036.174.255.247.163.027	20
44	8.486.324.855	10
44	72.017.709.544.590.771.027	20
43	3.881.830.257	10
43	15.068.606.144.160.686.051	20
42	2.496.801.518	10
42	6.234.017.820.287.104.326	19
41	1.950.914.241	10
41	3.806.066.375.736.606.083	19
40	1.912.253.831	10
40	3.656.714.714.174.176.563	19
39	1.340.374.437	10
39	1.796.603.631.363.066.971	19
38	813.448.358	9
38	661.698.231.132.896.166	18
37	620.208.559	9
37	384.658.656.656.856.483	18
36	452.357.920	9
36	204.627.687.786.726.402	18
35	451.977.320	9
35	204.283.497.794.382.402	18
34	246.553.448	9
34	60.788.602.720.688.706	17
33	186.062.629	9
33	34.619.301.910.391.643	17
32	79.094.282	8
32	6.255.905.445.095.526	16
31	61.875.091	8
31	3.828.526.886.258.283	16
30	28.559.254	8
30	815.630.989.036.518	15
29	26.237.622	8
29	688.412.808.214.886	15
28	25.809.892	8
28	666.150.525.051.666	15
27	14.365.940	8
27	206.380.232.083.602	15
26	5.513.951	7
26	30.403.655.630.403	14
25	2.491.032	7
25	6.205.240.425.026	13
24	1.845.571	7
24	3.406.132.316.043	13
23	1.050.503	7
23	1.103.556.553.011	13
22	902.696	6
22	814.860.068.418	12
21	423.437	6
21	179.298.892.971	12
20	298.436	6
20	89.064.046.098	11
19	268.865	6
19	72.288.388.227	11
18	140.833	6
18	19.833.933.891	11
17	10.503	5
17	110.313.011	9
16	4.540	4
16	20.611.602	8
15	3.627	4
15	13.155.131	8
14	2.986	4
14	8.916.198	7
13	2.916	4
13	8.503.058	7
12	2.538	4
12	6.441.446	7
11	1.811	4
11	3.279.723	7
10	828	3
10	685.586	6
9	393	2
9	154.451	6
8	258	2
8	66.566	5
7	85	2
7	7.227	4
6	19	2
6	363	3
5	13	2
5	171	3
4	8	1
4	66	2
3	3	1
3	11	2
2	2	1
2	6	1
1	1	1
1	3	1
0	0	1
0	2	1

	Case X = 3
One can find the regular numbers of the form n² + 3 at %N n^2 + 3. under A117950. The palindromic numbers of the form n² + 3 are categorised as follows : %N n^2 + 3 is a palindrome. under A0?????. %N Palindromes of the form n^2 + 3. under A0?????.
51	211.877.852.724.471	15
51	44.892.224.475.132.623.157.442.229.844	29
50	200.000.000.000.001	15
50	40.000.000.000.000.400.000.000.000.004	29
49	159.657.521.992.893	15
49	25.490.524.328.911.111.982.342.509.452	29
48	145.112.248.387.503	15
48	21.057.564.632.076.367.023.646.575.012	29
47	27.276.194.939.188	14
47	743.990.810.360.585.063.018.099.347	27
46	20.000.000.000.001	14
46	400.000.000.000.040.000.000.000.004	27
45	2.201.668.436.941	13
45	4.847.343.906.222.226.093.437.484	25
44	2.006.428.407.601	13
44	4.025.754.954.828.284.594.575.204	25
43	2.000.000.000.001	13
43	4.000.000.000.004.000.000.000.004	25
42	312.477.762.974	12
42	97.642.352.353.235.325.324.679	23
41	200.000.000.001	12
41	40.000.000.000.400.000.000.004	23
40	175.791.724.530	12
40	30.902.730.413.231.403.720.903	23
39	170.460.122.783	12
39	29.056.653.459.195.435.665.092	23
38	160.164.390.407	12
38	25.652.631.954.445.913.625.652	23
37	20.000.000.001	11
37	400.000.000.040.000.000.004	21
36	17.038.650.783	11
36	290.315.620.505.026.513.092	21
35	2.196.849.659	10
35	4.826.148.424.248.416.284	19
34	2.020.653.099	10
34	4.083.038.946.498.303.804	19
33	2.011.953.701	10
33	4.047.957.694.967.597.404	19
33	2.000.000.001	10
33	4.000.000.004.000.000.004	19
32	1.750.740.840	10
32	3.065.093.488.843.905.603	19
31	1.745.447.580	10
31	3.046.587.254.527.856.403	19
30	1.582.907.507	10
30	2.505.596.175.716.955.052	19
29	287.490.325	9
29	82.650.686.968.605.628	17
28	276.529.592	9
28	76.468.615.251.686.467	17
27	201.146.451	9
27	40.459.894.749.895.404	17
26	200.000.001	9
26	40.000.000.400.000.004	17
25	174.546.080	9
25	30.466.334.043.366.403	17
24	20.104.299	8
24	404.182.838.281.404	15
23	20.000.001	8
23	400.000.040.000.004	15
22	2.808.622	7
22	7.888.357.538.887	13
21	2.211.109	7
21	4.889.003.009.884	13
20	2.000.001	7
20	4.000.004.000.004	13
19	200.001	6
19	40.000.400.004	11
18	167.313	6
18	27.993.639.972	11
17	165.837	6
17	27.501.910.572	11
16	152.127	6
16	23.142.624.132	11
15	145.797	6
15	21.256.765.212	11
14	31.614	5
14	999.444.999	9
13	30.506	5
13	930.616.039	9
12	21.129	5
12	446.434.644	9
11	20.001	5
11	400.040.004	9
10	2.875	4
10	8.265.628	7
9	2.001	4
9	4.004.004	7
8	306	3
8	93.639	5
7	201	3
7	40.404	5
6	147	3
6	21.612	5
5	28	2
5	787	3
4	21	2
4	444	3
3	17	2
3	292	3
2	2	1
2	7	1
1	1	1
1	4	1
0	0	1
0	3	1

	Case X = 4
One can find the regular numbers of the form n² + 4 at %N n^2 + 4. under A087475. The palindromic numbers of the form n² + 4 are categorised as follows : %N n^2 + 4 is a palindrome. under A0?????. %N Palindromes of the form n^2 + 4. under A0?????.
64	290.001.098.900.012	15
64	84.100.637.363.214.541.236.373.600.148	29
63	290.000.000.000.012	15
63	84.100.000.000.006.960.000.000.000.148	29
62	225.116.991.118.801	15
62	50.677.659.690.382.328.309.695.677.605	29
61	200.000.000.000.000	15
61	40.000.000.000.000.000.000.000.000.004	29
60	29.001.099.890.012	14
60	841.063.794.830.454.038.497.360.148	27
59	29.001.089.109.988	14
59	841.063.169.565.464.565.961.360.148	27
58	29.000.108.900.012	14
58	841.006.316.212.555.212.613.600.148	27
57	29.000.010.999.988	14
57	841.000.637.999.424.999.736.000.148	27
56	29.000.000.000.012	14
56	841.000.000.000.696.000.000.000.148	27
55	24.096.294.452.009	14
55	580.631.406.317.919.713.604.136.085	27
54	23.283.121.322.421	14
54	542.103.738.514.575.415.837.301.245	27
53	20.102.019.998.520	14
53	404.091.208.020.898.020.802.190.404	27
52	20.000.000.000.000	14
52	400.000.000.000.000.000.000.000.004	27
51	17.784.936.509.397	14
51	316.303.966.643.282.346.669.303.613	27
50	2.900.109.890.012	13
50	8.410.637.374.145.414.737.360.148	25
49	2.900.000.000.012	13
49	8.410.000.000.069.600.000.000.148	25
48	2.000.000.000.000	13
48	4.000.000.000.000.000.000.000.004	25
47	1.893.180.648.207	13
47	3.584.132.966.745.476.692.314.853	25
46	291.098.901.088	12
46	84.738.570.214.641.207.583.748	23
45	290.010.890.012	12
45	84.106.316.325.552.361.360.148	23
44	290.001.099.988	12
44	84.100.637.994.249.973.600.148	23
43	290.000.000.012	12
43	84.100.000.006.960.000.000.148	23
42	234.304.304.471	12
42	54.898.507.093.639.070.589.845	23
41	200.000.000.000	12
41	40.000.000.000.000.000.000.004	23
40	29.010.989.012	11
40	841.637.483.454.384.736.148	21
39	29.000.000.012	11
39	841.000.000.696.000.000.148	21
38	20.046.487.910	11
38	401.861.677.525.776.168.104	21
37	20.000.000.000	11
37	400.000.000.000.000.000.004	21
36	2.901.089.012	10
36	8.416.317.455.547.136.148	19
35	2.900.109.988	10
35	8.410.637.942.497.360.148	19
34	2.900.000.012	10
34	8.410.000.069.600.000.148	19
33	2.418.272.559	10
33	5.848.042.169.612.408.485	19
32	2.000.000.000	10
32	4.000.000.000.000.000.004	19
31	1.981.904.483	10
31	3.927.945.379.735.497.293	19
30	293.695.292	9
30	86.256.924.542.965.268	17
29	291.098.912	9
29	84.738.576.567.583.748	17
28	290.000.012	9
28	84.100.006.960.000.148	17
27	200.000.000	9
27	40.000.000.000.000.004	17
26	197.605.717	9
26	39.048.019.391.084.093	17
25	177.437.653	9
25	31.484.120.702.148.413	17
24	29.108.912	8
24	847.328.757.823.748	15
23	29.010.988	8
23	841.637.424.736.148	15
22	29.000.012	8
22	841.000.696.000.148	15
21	24.208.291	8
21	586.041.353.140.685	15
20	23.020.911	8
20	529.962.343.269.925	15
19	20.000.000	8
19	400.000.000.000.004	15
18	2.900.012	7
18	8.410.069.600.148	13
17	2.000.000	7
17	4.000.000.000.004	13
16	291.088	6
16	84.732.223.748	11
15	290.012	6
15	84.106.960.148	11
14	284.298	6
14	80.825.352.808	11
13	200.000	6
13	40.000.000.004	11
12	29.012	5
12	841.696.148	9
11	20.230	5
11	409.252.904	9
10	20.000	5
10	400.000.004	9
9	2.912	4
9	8.479.748	7
8	2.369	4
8	5.612.165	7
7	2.329	4
7	5.424.245	7
6	2.000	4
6	4.000.004	7
5	241	3
5	58.085	5
4	200	3
4	40.004	5
3	20	2
3	404	3
2	2	1
2	8	1
1	1	1
1	5	1
0	0	1
0	4	1

	Case X = 5
One can find the regular numbers of the form n² + 5 at %N n^2 + 5. under A117951. The palindromic numbers of the form n² + 5 are categorised as follows : %N n^2 + 5 is a palindrome. under A0?????. %N Palindromes of the form n^2 + 5. under A0?????.
43	309.910.900.890.992	15
43	96.044.766.491.066.266.019.466.744.069	29
42	262.407.097.891.859	15
42	68.857.485.024.027.672.042.058.475.886	29
41	204.099.968.332.397	15
41	41.656.797.073.285.458.237.079.765.614	29
40	120.984.198.909.694	15
40	14.637.176.385.820.402.858.367.173.641	29
39	120.357.148.031.794	15
39	14.485.843.082.347.174.328.034.858.441	29
38	112.770.894.443.846	15
38	12.717.274.633.665.056.633.647.271.721	29
37	31.090.089.099.108	14
37	966.593.640.190.474.091.046.395.669	27
36	30.990.990.990.992	14
36	960.441.522.603.747.306.225.144.069	27
35	25.763.006.263.481	14
35	663.732.491.732.161.237.194.237.366	27
34	12.686.342.618.416	14
34	160.943.289.031.838.130.982.349.061	27
33	3.048.771.427.882	13
33	9.295.007.219.469.649.127.005.929	25
32	2.256.574.603.770	13
32	5.092.128.942.379.732.498.212.905	25
31	309.910.890.992	12
31	96.044.760.355.455.306.744.069	23
30	225.101.642.740	12
30	50.670.749.564.246.594.707.605	23
29	218.488.832.637	12
29	47.737.369.987.078.996.373.774	23
28	214.091.717.957	12
28	45.835.263.697.779.636.253.854	23
27	30.990.990.992	11
27	960.441.522.666.225.144.069	21
26	22.360.694.000	11
26	500.000.636.161.636.000.005	21
25	12.996.564.684	11
25	168.910.693.585.396.019.861	21
24	3.108.910.892	10
24	9.665.326.934.396.235.669	19
23	3.099.109.008	10
23	9.604.476.643.466.744.069	19
22	3.049.566.118	10
22	9.299.853.508.053.589.929	19
21	2.172.438.487	10
21	4.719.488.979.798.849.174	19
20	2.171.728.987	10
20	4.716.406.792.976.046.174	19
19	310.900.892	9
19	96.659.364.646.395.669	17
18	307.606.262	9
18	94.621.612.421.612.649	17
17	204.050.603	9
17	41.636.648.584.663.614	17
16	203.480.347	9
16	41.404.251.615.240.414	17
15	31.089.108	8
15	966.532.636.235.669	15
14	30.990.992	8
14	960.441.585.144.069	15
13	24.619.199	8
13	606.104.959.401.606	15
12	20.920.277	8
12	437.657.989.756.734	15
11	13.778.324	8
11	189.842.212.248.981	15
10	1.007.436	7
10	1.014.927.294.101	13
9	310.892	6
9	96.653.835.669	11
8	136.474	6
8	18.625.152.681	11
7	22.570	5
7	509.404.905	9
6	20.347	5
6	414.000.414	9
5	1.374	4
5	1.887.881	7
4	1.014	4
4	1.028.201	7
3	203	3
3	41.214	5
2	2	1
2	9	1
1	1	1
1	6	1
0	0	1
0	5	1

	Case X = 6
One can find the regular numbers of the form n² + 6 at %N n^2 + 6. under A114949. The palindromic numbers of the form n² + 6 are categorised as follows : %N n^2 + 6 is a palindrome. under A0?????. %N Palindromes of the form n^2 + 6. under A0?????.
45	455.835.912.156.236	15
45	207.786.378.811.307.703.118.873.687.702	30
44	114.028.867.547.995	15
44	13.002.582.634.278.187.243.628.520.031	29
43	27.668.521.984.581	14
43	765.547.108.811.242.118.801.745.567	27
42	1.622.457.383.434	13
42	2.632.367.961.059.501.697.632.362	25
41	360.616.362.245	12
41	130.044.160.718.817.061.440.031	24
40	234.732.340.243	12
40	55.099.271.555.955.517.299.055	23
39	162.829.334.666	12
39	26.513.392.227.772.229.331.562	23
38	157.495.523.406	12
38	24.804.839.892.929.893.840.842	23
37	157.302.301.856	12
37	24.744.014.169.196.141.044.742	23
36	144.260.002.214	12
36	20.810.948.238.783.284.901.802	23
35	36.131.148.255	11
35	1.305.459.874.224.789.545.031	22
34	28.043.323.459	11
34	786.427.990.626.099.724.687	21
33	27.979.618.659	11
33	782.859.060.303.060.958.287	21
32	3.645.161.865	10
32	13.287.205.022.050.278.231	20
31	2.354.704.143	10
31	5.544.631.601.061.364.455	19
30	1.554.076.056	10
30	2.415.152.387.832.515.142	19
29	360.581.005	9
29	130.018.661.166.810.031	18
28	156.832.744	9
28	24.596.509.590.569.542	17
27	144.402.286	9
27	20.852.020.202.025.802	17
26	88.574.691	8
26	7.845.475.885.745.487	16
25	24.589.420	8
25	604.639.575.936.406	15
24	23.954.037	8
24	573.795.888.597.375	15
23	16.963.924	8
23	287.774.717.477.782	15
22	11.504.365	8
22	132.350.414.053.231	15
21	11.493.715	8
21	132.105.484.501.231	15
20	8.493.489	7
20	72.139.355.393.127	14
19	7.586.663	7
19	57.557.455.475.575	14
18	245.820	6
18	60.427.472.406	11
17	230.623	6
17	53.186.968.135	11
16	169.874	6
16	28.857.175.882	11
15	144.664	6
15	20.927.672.902	11
14	86.379	5
14	7.461.331.647	10
13	16.776	5
13	281.434.182	9
12	15.556	5
12	241.989.142	9
11	11.515	5
11	132.595.231	9
10	2.729	4
10	7.447.447	7
9	156	3
9	24.342	5
8	115	3
8	13.231	5
7	73	2
7	5.335	4
6	23	2
6	535	3
5	16	2
5	262	3
4	14	2
4	202	3
3	7	1
3	55	2
2	4	1
2	22	2
1	1	1
1	7	1
0	0	1
0	6	1

	Case X = 7
One can find the regular numbers of the form n² + 7 at %N n^2 + 7. under A117619. The palindromic numbers of the form n² + 7 are categorised as follows : %N n^2 + 7 is a palindrome. under A0?????. %N Palindromes of the form n^2 + 7. under A0?????.
63	928.397.928.617.619	15
63	861.922.713.861.485.584.168.317.229.168	30
62	894.779.501.236.751	15
62	800.630.355.833.488.884.338.553.036.008	30
61	390.478.814.572.838	15
61	152.473.704.630.208.802.036.407.374.251	30
60	247.007.970.190.247	15
60	61.012.937.337.505.950.573.373.921.016	29
59	139.767.782.028.828	15
59	19.535.032.893.257.975.239.823.053.591	29
58	124.845.622.180.888	15
58	15.586.429.377.733.033.777.392.468.551	29
57	26.346.484.384.433	14
57	694.137.239.419.171.914.932.731.496	27
56	15.258.510.823.135	14
56	232.822.152.539.727.935.251.228.232	27
55	11.545.204.077.982	14
55	133.291.737.202.252.202.737.192.331	27
54	8.372.856.137.010	13
54	70.104.719.891.066.019.891.740.107	26
53	8.356.234.672.333	13
53	69.826.657.899.100.199.875.662.896	26
52	8.309.775.585.967	13
52	69.052.370.289.133.198.207.325.096	26
51	3.693.414.147.568	13
51	13.641.308.065.455.456.080.314.631	26
50	1.065.362.822.952	13
50	1.134.997.944.528.254.497.994.311	25
49	921.146.111.279	12
49	848.510.158.324.423.851.015.848	24
48	899.067.950.651	12
48	808.323.179.887.788.971.323.808	24
47	840.316.026.460	12
47	706.131.024.325.523.420.131.607	24
46	837.309.460.990	12
46	701.087.133.463.364.331.780.107	24
45	265.559.939.850	12
45	70.522.081.653.135.618.022.507	23
44	255.467.159.557	12
44	65.263.469.612.121.696.436.256	23
43	248.721.045.947	12
43	61.862.158.696.969.685.126.816	23
42	191.421.191.716	12
42	36.642.072.637.973.627.024.663	23
41	181.226.428.104	12
41	32.843.018.243.334.281.034.823	23
40	175.005.213.314	12
40	30.626.824.687.078.642.862.603	23
39	151.717.587.105	12
39	23.018.226.236.963.262.281.032	23
38	140.747.479.122	12
38	19.809.852.879.197.825.890.891	23
37	26.642.122.830	11
37	709.802.708.888.807.208.907	21
36	17.554.182.214	11
36	308.149.313.202.313.941.803	21
35	17.471.049.636	11
35	305.237.575.383.575.732.503	21
34	11.551.134.018	11
34	133.428.697.101.796.824.331	21
33	5.912.650.544	10
33	34.959.436.455.463.495.943	20
32	4.224.722.192	10
32	17.848.277.599.577.284.871	20
31	2.602.899.363	10
31	6.775.085.093.905.805.776	19
30	1.789.471.504	10
30	3.202.208.263.628.022.023	19
29	1.735.796.064	10
29	3.012.987.975.797.892.103	19
28	152.330.735	9
28	23.204.652.825.640.232	17
27	91.698.429	8
27	8.408.601.881.068.048	16
26	90.991.811	8
26	8.279.509.669.059.728	16
25	83.941.420	8
25	7.046.161.991.616.407	16
24	26.458.500	8
24	700.052.222.250.007	15
23	25.654.957	8
23	658.176.818.671.856	15
22	19.049.016	8
22	362.865.010.568.263	15
21	18.070.704	8
21	326.550.343.055.623	15
20	5.902.906	7
20	34.844.299.244.843	14
19	4.821.815	7
19	23.249.899.894.232	14
18	2.654.580	7
18	7.046.794.976.407	13
17	1.815.554	7
17	3.296.236.326.923	13
16	823.363	6
16	677.926.629.776	12
15	190.034	6
15	36.112.921.163	11
14	180.704	6
14	32.653.935.623	11
13	15.195	5
13	230.888.032	9
12	10.502	5
12	110.292.011	9
11	5.696	4
11	32.444.423	8
10	4.805	4
10	23.088.032	8
9	911	3
9	829.928	6
8	247	3
8	61.016	5
7	118	3
7	13.931	5
6	42	2
6	1.771	4
5	29	2
5	848	3
5	15	2
5	232	3
4	12	2
4	151	3
3	9	1
3	88	2
2	2	1
2	11	2
1	1	1
1	8	1
0	0	1
0	7	1

	Case X = 8
One can find the regular numbers of the form n² + 8 at %N n^2 + 8. under A189833. The palindromic numbers of the form n² + 8 are categorised as follows : %N n^2 + 8 is a palindrome. under A0?????. %N Palindromes of the form n^2 + 8. under A0?????.
75	576.376.983.170.335	15
75	332.210.426.728.536.635.827.624.012.233	30
74	481.209.766.450.682	15
74	231.562.839.327.519.915.723.938.265.132	30
73	300.000.000.000.001	15
73	90.000.000.000.000.600.000.000.000.009	29
72	275.337.945.056.293	15
72	75.810.983.987.822.222.878.938.901.857	29
71	216.065.920.101.284	15
71	46.684.481.829.214.441.292.818.448.664	29
70	94.896.997.097.501	14
70	9.005.440.058.123.113.218.500.445.009	28
69	46.837.097.362.148	14
69	2.193.713.689.311.331.139.863.173.912	28
68	30.000.000.000.001	14
68	900.000.000.000.060.000.000.000.009	27
67	28.285.433.123.400	14
67	800.065.726.978.333.879.627.560.008	27
66	27.189.036.465.673	14
66	739.243.703.931.696.139.307.342.937	27
65	22.099.529.046.626	14
65	488.389.184.082.666.280.481.983.884	27
64	21.644.889.670.816	14
64	468.501.248.861.797.168.842.105.864	27
63	16.531.889.324.942	14
63	273.303.364.652.131.256.463.303.372	27
62	8.970.276.377.420	13
62	80.465.858.287.299.278.285.856.408	26
61	5.438.671.922.172	13
61	29.579.152.277.022.077.225.197.592	26
60	5.003.245.383.362	13
60	25.032.464.366.133.166.346.423.052	26
59	3.079.298.219.029	13
59	9.482.077.521.715.171.257.702.849	25
58	3.000.000.000.001	13
58	9.000.000.000.006.000.000.000.009	25
57	1.462.512.223.952	13
57	2.138.942.005.209.025.002.498.312	25
56	313.136.321.259	12
56	98.054.355.691.619.655.345.089	23
55	300.000.000.001	12
55	90.000.000.000.600.000.000.009	23
54	275.014.002.593	12
54	75.632.701.622.222.610.723.657	23
53	270.510.768.373	12
53	73.176.075.805.750.857.067.137	23
52	267.374.937.347	12
52	71.489.357.121.312.175.398.417	23
51	64.869.123.996	11
51	4.208.003.248.008.423.008.024	22
50	30.000.000.001	11
50	900.000.000.060.000.000.009	21
49	27.157.572.473	11
49	737.533.742.626.247.335.737	21
48	18.229.113.115	11
48	332.300.564.959.465.003.233	21
47	16.573.604.592	11
47	274.684.369.171.963.486.472	21
46	9.607.915.389	10
46	92.312.038.122.183.021.329	20
45	5.764.428.665	10
45	33.228.637.833.873.682.233	20
44	4.813.848.682	10
44	23.173.139.133.193.137.132	20
43	3.000.000.001	10
43	9.000.000.006.000.000.009	19
42	2.193.798.276	10
42	4.812.750.875.780.572.184	19
41	1.660.759.758	10
41	2.758.122.973.792.218.572	19
40	984.531.531	9
40	969.302.335.533.203.969	18
39	899.936.670	9
39	809.886.010.010.688.908	18
38	858.762.723	9
38	737.473.414.414.374.737	18
37	300.000.001	9
37	90.000.000.600.000.009	17
36	268.086.303	9
36	71.870.265.856.207.817	17
35	167.274.158	9
35	27.980.643.934.608.972	17
34	153.532.282	9
34	23.572.161.616.127.532	17
33	30.000.001	8
33	900.000.060.000.009	15
32	28.370.280	8
32	804.872.787.278.408	15
31	21.969.674	8
31	482.666.575.666.284	15
30	14.573.302	8
30	212.381.131.183.212	15
29	3.000.001	7
29	9.000.006.000.009	13
28	2.837.650	7
28	8.052.257.522.508	13
27	2.198.624	7
27	4.833.947.493.384	13
26	2.050.504	7
26	4.204.566.654.024	13
25	576.285	6
25	332.104.401.233	12
24	306.921	6
24	94.200.500.249	11
23	300.001	6
23	90.000.600.009	11
22	283.620	6
22	80.440.304.408	11
21	96.211	5
21	9.256.556.529	10
20	85.773	5
20	7.357.007.537	10
19	30.001	5
19	900.060.009	9
18	27.543	5
18	758.616.857	9
17	20.504	5
17	420.414.024	9
16	18.235	5
16	332.515.233	9
15	8.707	4
15	75.811.857	8
14	6.364	4
14	40.500.504	8
13	3.001	4
13	9.006.009	7
12	2.840	4
12	8.065.608	7
11	951	3
11	904.409	6
10	654	3
10	427.724	6
9	301	3
9	90.609	5
8	216	3
8	46.664	5
7	172	3
7	29.592	5
6	148	3
6	21.912	5
5	31	2
5	969	3
4	27	2
4	737	3
3	6	1
3	44	2
2	5	1
2	33	2
1	1	1
1	9	1
0	0	1
0	8	1

	Case X = 9
One can find the regular numbers of the form n² + 9 at %N n^2 + 9. under A189834. The palindromic numbers of the form n² + 9 are categorised as follows : %N n^2 + 9 is a palindrome. under A0?????. %N Palindromes of the form n^2 + 9. under A0?????.
44	300.000.000.000.000	15
44	90.000.000.000.000.000.000.000.000.009	29
43	299.142.331.546.067	15
43	89.486.134.522.817.071.822.543.168.498	29
42	241.848.269.901.926	15
42	58.490.585.654.554.845.545.658.509.485	29
41	233.022.488.525.894	15
41	54.299.480.158.800.400.885.108.499.245	29
40	30.000.000.000.000	14
40	900.000.000.000.000.000.000.000.009	27
39	3.000.020.251.100	13
39	9.000.121.507.010.107.051.210.009	25
38	3.000.000.000.000	13
38	9.000.000.000.000.000.000.000.009	25
37	2.983.356.089.717	13
37	8.900.413.558.051.508.553.140.098	25
36	301.053.306.560	12
36	90.633.093.390.709.339.033.609	23
35	300.804.083.720	12
35	90.483.096.782.628.769.038.409	23
34	300.000.000.000	12
34	90.000.000.000.000.000.000.009	23
33	291.644.222.507	12
33	85.056.352.521.712.525.365.058	23
32	288.652.169.573	12
32	83.320.074.999.199.947.002.338	23
31	285.771.472.353	12
31	81.665.334.410.801.443.356.618	23
30	193.566.472.908	12
30	37.467.979.434.043.497.976.473	23
29	189.532.984.712	12
29	35.922.752.293.839.225.722.953	23
28	30.000.000.000	11
28	900.000.000.000.000.000.009	21
27	3.000.000.000	10
27	9.000.000.000.000.000.009	19
26	1.836.534.082	10
26	3.372.857.434.347.582.733	19
25	301.021.960	9
25	90.614.220.402.241.609	17
24	300.000.000	9
24	90.000.000.000.000.009	17
23	292.676.657	9
23	85.659.625.552.695.658	17
22	198.306.128	9
22	39.325.320.402.352.393	17
21	30.000.000	8
21	900.000.000.000.009	15
20	19.282.142	8
20	371.801.000.108.173	15
19	3.000.000	7
19	9.000.000.000.009	13
18	2.987.983	7
18	8.928.042.408.298	13
17	2.854.097	7
17	8.145.869.685.418	13
16	1.892.088	7
16	3.579.996.999.753	13
15	300.000	6
15	90.000.000.009	11
14	234.094	6
14	54.800.000.845	11
13	230.054	6
13	52.924.842.925	11
12	30.000	5
12	900.000.009	9
11	23.866	5
11	569.585.965	9
10	19.292	5
10	372.181.273	9
9	3.000	4
9	9.000.009	7
8	2.424	4
8	5.875.785	7
7	300	3
7	90.009	5
6	293	3
6	85.858	5
5	188	3
5	35.353	5
4	182	3
4	33.133	5
3	30	2
3	909	3
2	24	2
2	585	3
1	18	2
1	333	3
0	0	1
0	9	1

Further Topics Revealed

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