HOME plateWON | World!OfNumbers Palindromic Incremented Squaresof 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 { This 84... produces ever expanding Mandelbrot-like infinite palindromes }
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 BasenumberLength
Palindromic Incremented Squares of form n^2+XLength

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?????.
89815.731.447.843.771.74218
665.417.795.001.296.097.790.692.100.597.714.56636
88570.673.738.705.892.53918
325.668.516.048.561.312.213.165.840.615.866.52336
87555.231.465.460.388.59918
308.281.980.237.290.697.796.092.732.089.182.80336
86441.054.463.993.293.43318
194.529.040.208.411.373.373.114.802.040.925.49136
85292.076.758.112.830.56618
85.308.832.629.700.935.853.900.792.623.880.35835
84253.886.289.305.421.53818
64.458.247.897.276.202.820.267.279.874.285.44635
83250.562.271.637.470.58218
62.781.451.968.129.594.849.592.186.915.418.72635
82191.231.972.873.754.91918
36.569.667.449.188.537.173.588.194.476.696.56335
8185.000.416.618.604.76517
7.225.070.825.336.381.111.836.335.280.705.22734
8062.174.257.361.897.95917
3.865.638.278.503.522.552.253.058.728.365.68334
7929.281.657.627.282.73417
857.415.473.401.405.111.504.104.374.514.75833
7817.401.046.324.368.60117
302.796.413.182.821.999.128.281.314.697.20333
7714.325.552.760.917.15017
205.221.461.905.820.979.028.509.164.122.50233
7614.208.781.036.651.41017
201.889.458.547.504.717.405.745.854.988.10233
755.550.595.170.515.84916
30.809.106.746.953.866.835.964.760.190.80332
744.369.111.126.470.46716
19.089.132.035.448.033.084.453.023.198.09132
73Prime!    3.703.941.221.122.77716
13.719.180.569.532.488.423.596.508.191.73132
722.923.563.521.564.61616
8.547.223.664.623.298.923.264.663.227.45831
711.851.301.394.474.17116
3.427.316.853.182.010.102.813.586.137.24331
70817.683.802.207.30815
668.606.800.392.199.991.293.008.606.86630
69791.470.648.804.93215
626.425.787.919.700.007.919.787.524.62630
68364.768.101.292.42715
133.055.767.720.482.284.027.767.550.33130
67284.676.297.563.00415
81.040.594.394.179.997.149.349.504.01829
66186.797.158.329.47115
34.893.178.359.965.456.995.387.139.84329
65141.426.468.677.10015
20.001.446.042.474.747.424.064.410.00229
64Prime!    140.737.100.319.08315
19.806.931.406.223.632.260.413.960.89129
63133.805.135.860.43715
17.903.814.382.630.003.628.341.830.97129
6293.455.496.826.37614
8.733.929.887.064.774.607.889.293.37828
6155.349.225.724.44914
3.063.536.788.296.006.928.876.353.60328
6045.492.981.517.64014
2.069.611.367.364.334.637.631.169.60228
5914.363.881.876.06014
206.321.102.549.404.945.201.123.60227
5813.413.161.997.31314
179.912.914.766.161.667.419.219.97127
5710.902.121.823.94714
118.856.260.264.181.462.062.658.81127
569.114.951.055.99413
83.082.332.753.166.135.723.328.03826
552.925.054.461.66613
8.555.943.603.712.173.063.495.55825
542.688.021.576.51513
7.225.459.995.810.185.999.545.22725
531.954.611.231.09113
3.820.505.064.707.074.605.050.28325
521.861.509.029.37913
3.465.215.866.459.546.685.125.64325
51268.803.849.76512
72.255.509.648.484.690.555.22723
50250.793.696.73212
62.897.478.320.502.387.479.82623
49144.598.910.53012
20.908.844.926.462.944.880.90223
48142.019.320.31012
20.169.487.341.314.378.496.10223
47Prime!    11.645.196.02311
135.610.590.414.095.016.53121
468.487.412.69510
72.036.174.255.247.163.02720
458.486.324.85510
72.017.709.544.590.771.02720
443.881.830.25710
15.068.606.144.160.686.05120
432.496.801.51810
6.234.017.820.287.104.32619
421.950.914.24110
3.806.066.375.736.606.08319
411.912.253.83110
3.656.714.714.174.176.56319
401.340.374.43710
1.796.603.631.363.066.97119
39813.448.3589
661.698.231.132.896.16618
38620.208.5599
384.658.656.656.856.48318
37452.357.9209
204.627.687.786.726.40218
36451.977.3209
204.283.497.794.382.40218
35246.553.4489
60.788.602.720.688.70617
34186.062.6299
34.619.301.910.391.64317
3379.094.2828
6.255.905.445.095.52616
3261.875.0918
3.828.526.886.258.28316
3128.559.2548
815.630.989.036.51815
3026.237.6228
688.412.808.214.88615
2925.809.8928
666.150.525.051.66615
2814.365.9408
206.380.232.083.60215
275.513.9517
30.403.655.630.40314
262.491.0327
6.205.240.425.02613
251.845.5717
3.406.132.316.04313
24Prime!    1.050.5037
1.103.556.553.01113
23902.6966
814.860.068.41812
22423.4376
179.298.892.97112
21298.4366
89.064.046.09811
20268.8656
72.288.388.22711
19140.8336
19.833.933.89111
1810.5035
110.313.0119
174.5404
20.611.6028
163.6274
13.155.1318
152.9864
8.916.1987
142.9164
8.503.0587
132.5384
6.441.4467
12Prime!    1.8114
3.279.7237
118283
685.5866
103932
154.4516
92582
66.5665
8852
7.2274
7Prime!    192
3633
6Prime!    132
1713
581
662
4Prime!    31
Prime!    112
3Prime!    21
61
211
Prime!    31
101
Prime!    21

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?????.
65286.777.606.421.144.18518
82.241.395.544.640.678.487.604.644.559.314.22835
64265.887.849.560.347.59818
70.696.348.543.826.036.363.062.834.584.369.60735
63265.769.807.591.493.40218
70.633.590.627.219.423.932.491.272.609.533.60735
62200.000.000.000.000.00118
40.000.000.000.000.000.400.000.000.000.000.00435
61174.471.663.807.316.38018
30.440.361.471.693.232.423.239.617.416.304.40335
6030.194.108.938.222.95417
911.684.214.573.275.282.572.375.412.486.11933
5926.513.359.380.045.19817
702.958.225.615.430.686.034.516.522.859.20733
5820.000.000.000.000.00117
400.000.000.000.000.040.000.000.000.000.00433
572.734.460.134.529.91216
7.477.272.227.333.344.433.337.222.727.74731
562.000.000.000.000.00116
4.000.000.000.000.004.000.000.000.000.00431
551.522.999.175.814.12316
2.319.526.489.530.497.940.359.846.259.13231
541.520.515.952.206.12316
2.311.968.760.913.292.923.190.678.691.13231
53211.877.852.724.47115
44.892.224.475.132.623.157.442.229.84429
52200.000.000.000.00115
40.000.000.000.000.400.000.000.000.00429
51159.657.521.992.89315
25.490.524.328.911.111.982.342.509.45229
50145.112.248.387.50315
21.057.564.632.076.367.023.646.575.01229
4927.276.194.939.18814
743.990.810.360.585.063.018.099.34727
4820.000.000.000.00114
400.000.000.000.040.000.000.000.00427
472.201.668.436.94113
4.847.343.906.222.226.093.437.48425
462.006.428.407.60113
4.025.754.954.828.284.594.575.20425
452.000.000.000.00113
4.000.000.000.004.000.000.000.00425
44312.477.762.97412
Prime!    97.642.352.353.235.325.324.67923
43200.000.000.00112
40.000.000.000.400.000.000.00423
42175.791.724.53012
30.902.730.413.231.403.720.90323
41170.460.122.78312
29.056.653.459.195.435.665.09223
40160.164.390.40712
25.652.631.954.445.913.625.65223
3920.000.000.00111
400.000.000.040.000.000.00421
3817.038.650.78311
290.315.620.505.026.513.09221
372.196.849.65910
4.826.148.424.248.416.28419
362.020.653.09910
4.083.038.946.498.303.80419
352.011.953.70110
4.047.957.694.967.597.40419
342.000.000.00110
4.000.000.004.000.000.00419
331.750.740.84010
3.065.093.488.843.905.60319
321.745.447.58010
3.046.587.254.527.856.40319
311.582.907.50710
2.505.596.175.716.955.05219
30287.490.3259
82.650.686.968.605.62817
29276.529.5929
76.468.615.251.686.46717
28201.146.4519
40.459.894.749.895.40417
27200.000.0019
40.000.000.400.000.00417
26174.546.0809
30.466.334.043.366.40317
2520.104.2998
404.182.838.281.40415
2420.000.0018
400.000.040.000.00415
232.808.6227
7.888.357.538.88713
22Prime!    2.211.1097
4.889.003.009.88413
212.000.0017
4.000.004.000.00413
20200.0016
40.000.400.00411
19167.3136
27.993.639.97211
18165.8376
27.501.910.57211
17152.1276
23.142.624.13211
16145.7976
21.256.765.21211
1531.6145
999.444.9999
1430.5065
Prime!    930.616.0399
1321.1295
446.434.6449
1220.0015
400.040.0049
112.8754
8.265.6287
102.0014
4.004.0047
93063
93.6395
82013
40.4045
71473
21.6125
6282
Prime!    7873
5212
4443
4Prime!    172
2923
3Prime!    21
Prime!    71
211
41
101
Prime!    31

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?????.
86290.001.098.901.099.98818
84.100.637.363.845.576.667.554.836.373.600.14835
85290.000.109.989.000.01218
84.100.063.793.632.104.540.123.639.736.000.14835
84290.000.010.890.000.01218
84.100.006.316.200.125.552.100.261.360.000.14835
83290.000.001.099.999.98818
84.100.000.637.999.994.249.999.973.600.000.14835
82290.000.000.000.000.01218
84.100.000.000.000.006.960.000.000.000.000.14835
81284.408.918.191.650.29818
80.888.432.746.944.831.813.844.964.723.488.80835
80200.000.000.000.000.00018
40.000.000.000.000.000.000.000.000.000.000.00435
7930.367.729.387.366.71517
922.198.988.144.335.999.533.441.889.891.22933
7829.000.109.998.900.01217
841.006.379.948.300.454.003.849.973.600.14833
7729.000.010.989.000.01217
841.000.637.362.121.454.121.263.736.000.14833
7629.000.000.000.000.01217
841.000.000.000.000.696.000.000.000.000.14833
7520.000.000.000.000.00017
400.000.000.000.000.000.000.000.000.000.00433
7419.271.985.571.352.23717
Prime!    371.409.427.862.408.808.804.268.724.904.17333
732.900.109.890.109.98816
8.410.637.374.713.766.673.174.737.360.14831
722.900.108.910.890.01216
8.410.631.695.023.651.563.205.961.360.14831
712.900.010.998.900.01216
8.410.063.793.741.045.401.473.973.600.14831
702.900.010.891.099.98816
8.410.063.168.498.546.458.948.613.600.14831
692.900.001.089.000.01216
8.410.006.316.201.255.521.026.136.000.14831
682.900.000.109.999.98816
8.410.000.637.999.942.499.997.360.000.14831
672.900.000.000.000.01216
8.410.000.000.000.069.600.000.000.000.14831
662.000.000.000.000.00016
4.000.000.000.000.000.000.000.000.000.00431
65290.001.098.900.01215
84.100.637.363.214.541.236.373.600.14829
64290.000.000.000.01215
84.100.000.000.006.960.000.000.000.14829
63225.116.991.118.80115
50.677.659.690.382.328.309.695.677.60529
62200.000.000.000.00015
40.000.000.000.000.000.000.000.000.00429
6129.001.099.890.01214
841.063.794.830.454.038.497.360.14827
6029.001.089.109.98814
841.063.169.565.464.565.961.360.14827
5929.000.108.900.01214
841.006.316.212.555.212.613.600.14827
5829.000.010.999.98814
841.000.637.999.424.999.736.000.14827
5729.000.000.000.01214
841.000.000.000.696.000.000.000.14827
5624.096.294.452.00914
580.631.406.317.919.713.604.136.08527
5523.283.121.322.42114
542.103.738.514.575.415.837.301.24527
5420.102.019.998.52014
404.091.208.020.898.020.802.190.40427
5320.000.000.000.00014
400.000.000.000.000.000.000.000.00427
5217.784.936.509.39714
316.303.966.643.282.346.669.303.61327
512.900.109.890.01213
8.410.637.374.145.414.737.360.14825
502.900.000.000.01213
8.410.000.000.069.600.000.000.14825
492.000.000.000.00013
4.000.000.000.000.000.000.000.00425
481.893.180.648.20713
3.584.132.966.745.476.692.314.85325
47291.098.901.08812
84.738.570.214.641.207.583.74823
46290.010.890.01212
84.106.316.325.552.361.360.14823
45290.001.099.98812
84.100.637.994.249.973.600.14823
44290.000.000.01212
84.100.000.006.960.000.000.14823
43234.304.304.47112
54.898.507.093.639.070.589.84523
42200.000.000.00012
40.000.000.000.000.000.000.00423
4129.010.989.01211
841.637.483.454.384.736.14821
4029.000.000.01211
841.000.000.696.000.000.14821
3920.046.487.91011
401.861.677.525.776.168.10421
3820.000.000.00011
400.000.000.000.000.000.00421
372.901.089.01210
8.416.317.455.547.136.14819
362.900.109.98810
8.410.637.942.497.360.14819
352.900.000.01210
8.410.000.069.600.000.14819
342.418.272.55910
5.848.042.169.612.408.48519
332.000.000.00010
4.000.000.000.000.000.00419
321.981.904.48310
Prime!    3.927.945.379.735.497.29319
31293.695.2929
86.256.924.542.965.26817
30291.098.9129
84.738.576.567.583.74817
29290.000.0129
84.100.006.960.000.14817
28200.000.0009
40.000.000.000.000.00417
27Prime!    197.605.7179
39.048.019.391.084.09317
26177.437.6539
31.484.120.702.148.41317
2529.108.9128
847.328.757.823.74815
2429.010.9888
841.637.424.736.14815
2329.000.0128
841.000.696.000.14815
2224.208.2918
586.041.353.140.68515
2123.020.9118
529.962.343.269.92515
2020.000.0008
400.000.000.000.00415
192.900.0127
8.410.069.600.14813
182.000.0007
4.000.000.000.00413
17291.0886
84.732.223.74811
16290.0126
84.106.960.14811
15284.2986
80.825.352.80811
14200.0006
40.000.000.00411
1329.0125
841.696.1489
1220.2305
409.252.9049
1120.0005
400.000.0049
102.9124
8.479.7487
92.3694
5.612.1657
82.3294
5.424.2457
72.0004
4.000.0047
6Prime!    2413
58.0855
52003
40.0045
4202
4043
3Prime!    21
81
211
Prime!    51
101
41

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?????.
57311.023.143.446.721.95818
96.735.395.759.480.184.048.108.495.759.353.76935
5622514845711723986018
50.691.827.742.273.596.169.537.224.772.819.60535
55222.757.722.839.648.61718
49.621.003.084.705.711.111.750.748.030.012.69435
5430.165.492.872.565.64817
909.956.960.244.808.909.808.442.069.659.90933
5312.873.477.448.137.33417
165.726.421.607.700.525.007.706.124.627.56133
5210.337.004.802.295.31417
106.853.668.282.676.383.676.282.866.358.60133
512.608.645.939.541.25916
6.805.033.637.885.097.905.887.363.305.08631
502.505.030.843.903.93916
6.275.179.528.910.080.800.198.259.715.72631
492.245.985.885.884.62016
5.044.452.599.592.921.292.959.952.544.40531
482.215.491.736.254.53316
4.908.403.633.412.125.212.143.363.048.09431
471.279.111.244.371.43416
1.636.125.575.477.438.347.745.755.216.36131
461.128.636.081.957.65416
1.273.819.405.496.724.276.945.049.183.72131
451.108.041.335.154.09616
1.227.755.600.410.071.700.140.065.577.22131
44309.910.900.890.99215
96.044.766.491.066.266.019.466.744.06929
43262.407.097.891.85915
68.857.485.024.027.672.042.058.475.88629
42204.099.968.332.39715
41.656.797.073.285.458.237.079.765.61429
41120.984.198.909.69415
14.637.176.385.820.402.858.367.173.64129
40120.357.148.031.79415
14.485.843.082.347.174.328.034.858.44129
39112.770.894.443.84615
12.717.274.633.665.056.633.647.271.72129
3831.090.089.099.10814
966.593.640.190.474.091.046.395.66927
3730.990.990.990.99214
960.441.522.603.747.306.225.144.06927
3625.763.006.263.48114
663.732.491.732.161.237.194.237.36627
3512.686.342.618.41614
160.943.289.031.838.130.982.349.06127
343.048.771.427.88213
9.295.007.219.469.649.127.005.92925
332.256.574.603.77013
5.092.128.942.379.732.498.212.90525
32309.910.890.99212
96.044.760.355.455.306.744.06923
31225.101.642.74012
50.670.749.564.246.594.707.60523
30218.488.832.63712
47.737.369.987.078.996.373.77423
29214.091.717.95712
45.835.263.697.779.636.253.85423
2830.990.990.99211
960.441.522.666.225.144.06921
2722.360.694.00011
500.000.636.161.636.000.00521
2612.996.564.68411
Prime!    168.910.693.585.396.019.86121
253.108.910.89210
9.665.326.934.396.235.66919
243.099.109.00810
9.604.476.643.466.744.06919
233.049.566.11810
9.299.853.508.053.589.92919
222.172.438.48710
4.719.488.979.798.849.17419
21Prime!    2.171.728.98710
4.716.406.792.976.046.17419
20310.900.8929
96.659.364.646.395.66917
19307.606.2629
94.621.612.421.612.64917
18204.050.6039
41.636.648.584.663.61417
17203.480.3479
41.404.251.615.240.41417
1631.089.1088
966.532.636.235.66915
1530.990.9928
960.441.585.144.06915
1424.619.1998
606.104.959.401.60615
1320.920.2778
437.657.989.756.73415
1213.778.3248
189.842.212.248.98115
111.007.4367
1.014.927.294.10113
10310.8926
96.653.835.66911
9136.4746
18.625.152.68111
822.5705
509.404.9059
7Prime!    20.3475
414.000.4149
61.3744
1.887.8817
51.0144
Prime!    1.028.2017
42033
41.2145
3Prime!    21
91
211
61
101
Prime!    51

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?????.
60518.382.838.739.954.31618
268.720.767.500.093.481.184.390.005.767.027.86236
59453.688.870.735.185.13618
205.833.591.428.967.527.725.769.824.195.338.50236
58Prime!    275.918.682.887.508.13118
76.131.119.566.377.272.327.277.366.591.113.16735
57272.826.851.328.266.52118
74.434.490.805.696.043.534.069.650.809.443.44735
56230.953.456.951.103.17718
53.339.499.277.665.068.386.056.677.299.493.33535
55150.456.252.387.445.79618
22.637.083.882.474.788.788.747.428.838.073.62235
54114.180.351.500.147.05518
13.037.152.668.697.133.833.179.686.625.173.03135
5324.683.119.453.972.73017
609.256.385.979.087.040.780.979.583.652.90633
5224.389.266.701.248.93317
594.836.330.224.650.010.056.422.033.638.49533
5123.622.123.718.428.45717
558.004.728.968.740.272.047.869.827.400.85533
5016.142.690.249.866.41617
260.586.448.503.132.252.231.305.844.685.06233
4914.480.123.530.112.33617
209.673.977.447.312.939.213.744.779.376.90233
482.773.554.286.001.03116
7.692.603.377.394.688.864.937.733.062.96731
471.140.454.064.936.00516
1.300.635.474.229.057.509.224.745.360.03131
46455.835.912.156.23615
207.786.378.811.307.703.118.873.687.70230
45114.028.867.547.99515
13.002.582.634.278.187.243.628.520.03129
4427.668.521.984.58114
765.547.108.811.242.118.801.745.56727
431.622.457.383.43413
2.632.367.961.059.501.697.632.36225
42360.616.362.24512
130.044.160.718.817.061.440.03124
41234.732.340.24312
55.099.271.555.955.517.299.05523
40162.829.334.66612
26.513.392.227.772.229.331.56223
39157.495.523.40612
24.804.839.892.929.893.840.84223
38157.302.301.85612
24.744.014.169.196.141.044.74223
37144.260.002.21412
20.810.948.238.783.284.901.80223
3636.131.148.25511
1.305.459.874.224.789.545.03122
3528.043.323.45911
Prime!    786.427.990.626.099.724.68721
3427.979.618.65911
782.859.060.303.060.958.28721
333.645.161.86510
13.287.205.022.050.278.23120
322.354.704.14310
5.544.631.601.061.364.45519
311.554.076.05610
2.415.152.387.832.515.14219
30360.581.0059
130.018.661.166.810.03118
29156.832.7449
24.596.509.590.569.54217
28144.402.2869
20.852.020.202.025.80217
2788.574.6918
7.845.475.885.745.48716
2624.589.4208
604.639.575.936.40615
2523.954.0378
573.795.888.597.37515
2416.963.9248
287.774.717.477.78215
2311.504.3658
132.350.414.053.23115
2211.493.7158
Prime!    132.105.484.501.23115
218.493.4897
72.139.355.393.12714
207.586.6637
57.557.455.475.57514
19245.8206
60.427.472.40611
18230.6236
53.186.968.13511
17169.8746
28.857.175.88211
16144.6646
20.927.672.90211
1586.3795
7.461.331.64710
1416.7765
281.434.1829
1315.5565
241.989.1429
1211.5155
Prime!    132.595.2319
11Prime!    2.7294
7.447.4477
101563
24.3425
91153
13.2315
8Prime!    732
5.3354
7Prime!    232
5353
6162
2623
5142
2023
4Prime!    71
552
341
222
211
Prime!    71
101
61

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?????.
80621.027.581.747.737.97618
385.675.257.291.443.374.473.344.192.752.576.58336
79265.381.792.368.454.18018
70.427.495.720.693.325.252.339.602.759.472.40735
78186.352.346.145.995.85618
34.727.196.914.117.056.565.071.141.969.172.74335
77175.384.862.900.595.23618
30.759.850.134.660.587.178.506.643.105.895.70335
76151.732.545.338.808.04518
23.022.765.314.993.438.983.439.941.356.722.03235
75123.558.432.548.360.16218
15.266.686.253.807.667.876.670.835.268.666.25135
74109.507.179.675.599.35218
11.991.822.400.503.999.799.930.500.422.819.91135
7382.414.649.436.802.81317
6.792.174.441.791.102.222.011.971.444.712.97634
7234.473.342.592.827.20217
1.188.411.349.522.434.114.342.259.431.148.81134
716.044.398.343.213.16616
36.534.751.331.438.066.083.413.315.743.56332
706.020.132.259.989.51616
36.241.992.427.766.477.466.772.429.914.26332
694.798.837.268.824.20516
23.028.839.132.656.155.165.623.193.882.03232
683.896.993.565.142.98816
15.186.558.846.765.855.856.764.885.568.15132
672.880.426.541.419.41116
8.296.857.060.513.389.833.150.607.586.92831
662.872.658.932.455.78916
8.252.169.342.218.033.308.122.439.612.52831
65928.397.928.617.61915
861.922.713.861.485.584.168.317.229.16830
64894.779.501.236.75115
800.630.355.833.488.884.338.553.036.00830
63390.478.814.572.83815
152.473.704.630.208.802.036.407.374.25130
62247.007.970.190.24715
61.012.937.337.505.950.573.373.921.01629
61139.767.782.028.82815
19.535.032.893.257.975.239.823.053.59129
60124.845.622.180.88815
15.586.429.377.733.033.777.392.468.55129
5926.346.484.384.43314
694.137.239.419.171.914.932.731.49627
5815.258.510.823.13514
232.822.152.539.727.935.251.228.23227
5711.545.204.077.98214
133.291.737.202.252.202.737.192.33127
568.372.856.137.01013
70.104.719.891.066.019.891.740.10726
558.356.234.672.33313
69.826.657.899.100.199.875.662.89626
548.309.775.585.96713
69.052.370.289.133.198.207.325.09626
533.693.414.147.56813
13.641.308.065.455.456.080.314.63126
521.065.362.822.95213
1.134.997.944.528.254.497.994.31125
51921.146.111.27912
848.510.158.324.423.851.015.84824
50899.067.950.65112
808.323.179.887.788.971.323.80824
49840.316.026.46012
706.131.024.325.523.420.131.60724
48837.309.460.99012
701.087.133.463.364.331.780.10724
47265.559.939.85012
70.522.081.653.135.618.022.50723
46255.467.159.55712
65.263.469.612.121.696.436.25623
45248.721.045.94712
61.862.158.696.969.685.126.81623
44191.421.191.71612
36.642.072.637.973.627.024.66323
43181.226.428.10412
32.843.018.243.334.281.034.82323
42175.005.213.31412
30.626.824.687.078.642.862.60323
41151.717.587.10512
23.018.226.236.963.262.281.03223
40140.747.479.12212
19.809.852.879.197.825.890.89123
3926.642.122.83011
709.802.708.888.807.208.90721
3817.554.182.21411
308.149.313.202.313.941.80321
3717.471.049.63611
305.237.575.383.575.732.50321
3611.551.134.01811
133.428.697.101.796.824.33121
355.912.650.54410
34.959.436.455.463.495.94320
344.224.722.19210
17.848.277.599.577.284.87120
332.602.899.36310
6.775.085.093.905.805.77619
321.789.471.50410
3.202.208.263.628.022.02319
311.735.796.06410
3.012.987.975.797.892.10319
30152.330.7359
23.204.652.825.640.23217
2991.698.4298
8.408.601.881.068.04816
2890.991.8118
8.279.509.669.059.72816
2783.941.4208
7.046.161.991.616.40716
2626.458.5008
700.052.222.250.00715
2525.654.9578
658.176.818.671.85615
2419.049.0168
362.865.010.568.26315
2318.070.7048
326.550.343.055.62315
225.902.9067
34.844.299.244.84314
214.821.8157
23.249.899.894.23214
202.654.5807
Prime!    7.046.794.976.40713
191.815.5547
3.296.236.326.92313
18823.3636
677.926.629.77612
17190.0346
36.112.921.16311
16180.7046
32.653.935.62311
1515.1955
230.888.0329
1410.5025
110.292.0119
135.6964
32.444.4238
124.8054
23.088.0328
11Prime!    9113
829.9286
102473
61.0165
91183
Prime!    13.9315
8422
1.7714
7Prime!    292
8483
6152
2323
5122
Prime!    1513
491
882
3Prime!    21
Prime!    112
211
81
101
Prime!    71

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?????.
84899.889.741.588.250.33018
809.801.547.015.767.955.559.767.510.745.108.90836
83300.000.000.000.000.00118
90.000.000.000.000.000.600.000.000.000.000.00935
82266.496.356.130.217.00318
71.020.307.830.683.449.594.438.603.870.302.01735
81161.066.420.577.345.21218
25.942.391.837.598.253.135.289.573.819.324.95235
80154.629.361.321.495.61818
23.910.239.382.693.645.054.639.628.393.201.93235
7930.000.000.000.000.00117
900.000.000.000.000.060.000.000.000.000.00933
786.513.071.537.407.80416
42.420.100.851.391.655.619.315.800.102.42432
773.000.000.000.000.00116
9.000.000.000.000.006.000.000.000.000.00931
76576.376.983.170.33515
332.210.426.728.536.635.827.624.012.23330
75481.209.766.450.68215
231.562.839.327.519.915.723.938.265.13230
74300.000.000.000.00115
90.000.000.000.000.600.000.000.000.00929
73275.337.945.056.29315
75.810.983.987.822.222.878.938.901.85729
72216.065.920.101.28415
46.684.481.829.214.441.292.818.448.66429
7194.896.997.097.50114
9.005.440.058.123.113.218.500.445.00928
7046.837.097.362.14814
2.193.713.689.311.331.139.863.173.91228
6930.000.000.000.00114
900.000.000.000.060.000.000.000.00927
6828.285.433.123.40014
800.065.726.978.333.879.627.560.00827
6727.189.036.465.67314
739.243.703.931.696.139.307.342.93727
6622.099.529.046.62614
488.389.184.082.666.280.481.983.88427
6521.644.889.670.81614
468.501.248.861.797.168.842.105.86427
6416.531.889.324.94214
273.303.364.652.131.256.463.303.37227
638.970.276.377.42013
80.465.858.287.299.278.285.856.40826
625.438.671.922.17213
29.579.152.277.022.077.225.197.59226
615.003.245.383.36213
25.032.464.366.133.166.346.423.05226
60Prime!    3.079.298.219.02913
9.482.077.521.715.171.257.702.84925
593.000.000.000.00113
9.000.000.000.006.000.000.000.00925
581.462.512.223.95213
2.138.942.005.209.025.002.498.31225
57313.136.321.25912
98.054.355.691.619.655.345.08923
56300.000.000.00112
90.000.000.000.600.000.000.00923
55275.014.002.59312
75.632.701.622.222.610.723.65723
54Prime!    270.510.768.37312
73.176.075.805.750.857.067.13723
53267.374.937.34712
71.489.357.121.312.175.398.41723
5264.869.123.99611
4.208.003.248.008.423.008.02422
51Prime!    30.000.000.00111
900.000.000.060.000.000.00921
50Prime!    27.157.572.47311
737.533.742.626.247.335.73721
4918.229.113.11511
332.300.564.959.465.003.23321
4816.573.604.59211
274.684.369.171.963.486.47221
479.607.915.38910
92.312.038.122.183.021.32920
465.764.428.66510
33.228.637.833.873.682.23320
454.813.848.68210
23.173.139.133.193.137.13220
443.000.000.00110
9.000.000.006.000.000.00919
432.193.798.27610
4.812.750.875.780.572.18419
421.660.759.75810
2.758.122.973.792.218.57219
41984.531.5319
969.302.335.533.203.96918
40899.936.6709
809.886.010.010.688.90818
39858.762.7239
737.473.414.414.374.73718
38300.000.0019
90.000.000.600.000.00917
37268.086.3039
71.870.265.856.207.81717
36167.274.1589
27.980.643.934.608.97217
35153.532.2829
23.572.161.616.127.53217
34Prime!    30.000.0018
900.000.060.000.00915
3328.370.2808
804.872.787.278.40815
3221.969.6748
482.666.575.666.28415
3114.573.3028
212.381.131.183.21215
303.000.0017
9.000.006.000.00913
292.837.6507
8.052.257.522.50813
282.198.6247
4.833.947.493.38413
272.050.5047
4.204.566.654.02413
26576.2856
332.104.401.23312
25306.9216
94.200.500.24911
24300.0016
90.000.600.00911
23283.6206
80.440.304.40811
22Prime!    96.2115
9.256.556.52910
2185.7735
7.357.007.53710
2030.0015
900.060.0099
1927.5435
758.616.8579
1820.5045
420.414.0249
1718.2355
332.515.2339
16Prime!    8.7074
75.811.8578
156.3644
40.500.5048
14Prime!    3.0014
9.006.0097
132.8404
8.065.6087
129513
904.4096
116543
427.7246
103013
90.6095
92163
46.6645
81723
29.5925
71483
21.9125
6Prime!    312
9693
5272
7373
461
442
3Prime!    51
332
211
91
101
81

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?????.
57300.015.673.824.225.70018
90.009.404.540.204.185.858.140.204.540.490.00935
56300.000.000.000.000.00018
90.000.000.000.000.000.000.000.000.000.000.00935
5530.074.841.278.007.12017
904.496.077.897.320.939.023.798.770.694.40933
5430.000.000.000.000.00017
900.000.000.000.000.000.000.000.000.000.00933
5324.213.184.844.776.47417
586.278.320.327.313.121.313.723.023.872.68533
5223.809.076.639.223.78417
566.872.130.412.431.717.134.214.031.278.66533
5123.724.561.255.743.48417
562.854.806.777.524.838.425.777.608.458.26533
5020.752.370.363.370.60517
430.660.875.698.502.616.205.896.578.066.03433
493.000.000.000.000.00016
9.000.000.000.000.000.000.000.000.000.00931
482.338.836.516.042.64416
5.470.156.248.774.492.944.778.426.510.74531
472.079.319.170.475.63516
4.323.568.212.707.482.847.072.128.653.23431
461.985.156.016.920.02216
3.940.844.411.513.766.673.151.144.480.49331
45300.000.000.000.00015
90.000.000.000.000.000.000.000.000.00929
44Prime!    299.142.331.546.06715
89.486.134.522.817.071.822.543.168.49829
43241.848.269.901.92615
58.490.585.654.554.845.545.658.509.48529
42233.022.488.525.89415
54.299.480.158.800.400.885.108.499.24529
4130.000.000.000.00014
900.000.000.000.000.000.000.000.00927
403.000.020.251.10013
9.000.121.507.010.107.051.210.00925
393.000.000.000.00013
9.000.000.000.000.000.000.000.00925
382.983.356.089.71713
8.900.413.558.051.508.553.140.09825
37301.053.306.56012
90.633.093.390.709.339.033.60923
36300.804.083.72012
90.483.096.782.628.769.038.40923
35300.000.000.00012
90.000.000.000.000.000.000.00923
34291.644.222.50712
85.056.352.521.712.525.365.05823
33288.652.169.57312
83.320.074.999.199.947.002.33823
32285.771.472.35312
81.665.334.410.801.443.356.61823
31193.566.472.90812
37.467.979.434.043.497.976.47323
30189.532.984.71212
35.922.752.293.839.225.722.95323
2930.000.000.00011
900.000.000.000.000.000.00921
283.000.000.00010
9.000.000.000.000.000.00919
271.836.534.08210
3.372.857.434.347.582.73319
26301.021.9609
Prime!    90.614.220.402.241.60917
25300.000.0009
90.000.000.000.000.00917
24292.676.6579
85.659.625.552.695.65817
23198.306.1289
39.325.320.402.352.39317
2230.000.0008
900.000.000.000.00915
2119.282.1428
371.801.000.108.17315
203.000.0007
9.000.000.000.00913
19Prime!    2.987.9837
8.928.042.408.29813
182.854.0977
8.145.869.685.41813
171.892.0887
3.579.996.999.75313
16300.0006
90.000.000.00911
15234.0946
54.800.000.84511
14230.0546
52.924.842.92511
1330.0005
900.000.0099
1223.8665
569.585.9659
1119.2925
372.181.2739
103.0004
9.000.0097
92.4244
5.875.7857
83003
90.0095
7Prime!    2933
85.8585
61883
Prime!    35.3535
51823
33.1335
4302
9093
3242
5853
2182
3333
101
91

Further Topics Revealed

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