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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

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

base x ( 3 x base - 2 )

 PLAIN TEXT POLYGONS

Normal and Palindromic Octagonals

So far this compilation counts 46 Palindromic Octagonals.
Here is the largest Sporadic Palindromic Octagonal that Patrick De Geest
discovered, using CUDA code by Robert Xiao, on [ June 16, 2023 ]

 This basenumber 1.505.189.034.316.273.989.876.364.558 has 28 digits yielding the following palindromic octagonal number 6.796.782.087.077.872.316.108.073.397.933.708.016.132.787.707.802.876.976 with a length of 55 digits.

A palindromic octagonal numbers can only end with digit 0, 1, 3, 5, 6 and 8.
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
3 can only be followed by 3 : 33
5 can only be followed by an even number : 50, 52, 54, 56 or 58
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
8 can only be followed by 0 : 80

There exist no palindromic octagonals of length
2, 3, 5, 6, 7, 8, 10, 11, 15, 17, 18, 20, 22, 23, 26, 31, 34, 38, 39, 40, 44, 45, 46, 47, 50, 52.
(Sloane's A059870)

 Case Octa Change of variables CUDApalin parameters Base Correction base n = m + 1 `A B C →  3 4 1 ` base = CUDAbase + 1

Sloane's A046183 gives the first numbers that are both Octagonal and Triangular.
1, 21, 11781, 203841, 113123361, ...
Consult also Eric Weisstein's page Octagonal Triangular Number.

Sloane's A036428 gives the first numbers that are both Octagonal and Square.
1, 225, 43681, 8473921, 1643897025, ...
Consult also Eric Weisstein's page Octagonal Square Number.

Sloane's A046189 gives the first numbers that are both Octagonal and Pentagonal.
1, 176, 1575425, 234631320, 2098015778145, ...
Consult also Eric Weisstein's page Octagonal Pentagonal Number.

Sloane's A046192 gives the first numbers that are both Octagonal and Hexagonal.
1, 11781, 113123361, 1086210502741, ...
Consult also Eric Weisstein's page Octagonal Hexagonal Number.

Sloane's A048906 gives the first numbers that are both Octagonal and Heptagonal.
1, 297045, 69010153345, ...
Consult also Eric Weisstein's page Octagonal Heptagonal Number.

Sloane's A048924 gives the first numbers that are both Octagonal and Nonagonal.
1, 631125, 286703855361, 130242107189808901, ...
Consult also Eric Weisstein's page Octagonal Nonagonal Number.

The best way to get a 'structural' insight as how to imagine octagonals is to visit for instance this site :

Sources Revealed

 Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences One can find the regular octagonal numbers at %N Octagonal numbers: n(3n–2) under A000567. The palindromic octagonal numbers are categorised as follows : %N n(3n–2) is a palindromic octagonal number under A057106. %N Palindromic octagonal numbers under A057107. Click here to view some of the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

The Table

Index NrInfoBasenumberLength
Palindromic OctagonalsLength

[PG8] Formula = n(3n–2)
46Info1.505.189.034.316.273.989.876.364.55828
6.796.782.087.077.872.316.108.073.397.933.708.016.132.787.707.802.876.97655
45InfoPrime!    1.050.527.056.977.712.014.268.976.32728
3.310.821.292.326.758.954.662.668.504.058.662.664.598.576.232.921.280.13355
44Info233.009.668.152.040.466.112.406.34327
162.880.516.356.972.063.257.851.225.522.158.752.360.279.653.615.088.26154
43Info134.655.235.447.046.729.292.188.66527
54.396.097.299.898.769.932.478.624.342.687.423.996.789.899.279.069.34553
42Info10.504.193.933.024.178.843.891.50726
331.014.270.547.745.901.058.874.464.478.850.109.547.745.072.410.13351
41Info1.505.231.393.963.863.016.048.05825
6.797.164.648.123.182.571.630.290.920.361.752.813.218.464.617.97649
40Info1.498.828.240.862.001.472.469.37625
6.739.458.286.816.445.701.381.236.321.831.075.446.186.828.549.37649
39Info1.057.171.651.503.698.963.468.08725
3.352.835.702.229.174.992.660.413.140.662.994.719.222.075.382.53349
38Info516.618.664.116.856.556.011.48224
800.684.532.341.656.355.118.723.327.811.553.656.143.235.486.00848
37Info1.061.445.921.245.492.013.31722
3.380.002.331.186.073.700.219.120.073.706.811.332.000.83343
36Info790.350.914.396.051.463.77121
1.873.963.703.660.024.006.763.676.004.200.663.073.693.78143
35Info736.230.589.620.506.244.34321
1.626.106.443.278.874.830.658.560.384.788.723.446.016.26143
34Info332.369.949.023.486.562.72721
331.409.349.041.625.168.935.539.861.526.140.943.904.13342
33Info246.919.023.530.449.916.31321
182.907.012.543.692.638.362.263.836.296.345.210.709.28142
32Info65.621.182.548.683.007.15320
12.918.418.797.262.737.739.193.773.726.279.781.481.92141
31Info704.687.633.763.271.76118
1.489.753.983.536.637.090.907.366.353.893.579.84137
30Info589.588.144.333.920.23318
1.042.842.539.817.346.669.666.437.189.352.482.40137
29Info221.420.450.951.946.11118
147.081.048.299.289.519.915.982.992.840.180.74136
28Info145.815.635.485.061.31818
63.786.598.655.736.820.002.863.755.689.568.73635
27Info136.626.014.502.187.50518
56.000.003.516.255.851.015.855.261.530.000.06535
26Info7.897.699.388.697.56316
187.120.966.902.701.565.107.209.669.021.78133
25Info4.731.413.168.091.25816
67.158.811.701.562.055.026.510.711.885.17632
24Info222.032.718.762.72315
147.895.584.603.498.894.306.485.598.74130
23Info105.802.560.494.38715
33.582.545.421.505.050.512.454.528.53329
22Info78.849.864.240.62114
18.651.903.272.292.929.227.230.915.68129
21Info51.722.791.547.84214
8.025.741.496.504.444.056.941.475.20828
20Info5.847.307.263.80113
102.573.006.711.888.117.600.375.20127
19Info1.400.295.262.09513
5.882.480.463.134.313.640.842.88525
18Info335.038.979.07712
336.753.352.502.205.253.357.63324
17Info13.175.129.92511
520.752.145.595.541.257.02521
16Info1.050.553.50710
3.310.988.011.108.890.13319
15Info42.687.0158
5.466.543.663.456.64516
14Info2.017.6317
12.212.500.521.22114
13Info1.472.7467
6.506.939.396.05613
12Info1.054.1677
3.333.802.083.33313
11Info1.054.0677
3.333.169.613.33313
10Info453.4866
616.947.749.61612
9Info426.1156
544.721.127.44512
8Info7.8634
185.464.5819
7Info7.3414
161.656.1619
6Info6.4314
124.060.4219
5Info6.3314
120.232.0219
4Info522
8.0084
3InfoPrime!    21
81
2Info11
11
1Info01
01

Contributions

The Lowell Family (email) submitted the first 8 palindromic octagonals.

Feng Yuan (email) from Washington State, USA, discovered the palindromic octagonals
starting from index number [18] up to [31].

[ January 3, 2008 ]
Feng Yuan (email) from Washington State, USA, discovered the palindromic octagonals
with index numbers [32] and [36].

Exhaustive search performed up to length 43 by Patrick De Geest and found [37] on [ July 17, 2022 ].

On [ July 5, 2022 ] PDG discovered [35] 736230589620506244343 which is smaller than Feng Yuan's record.
On [ July 30, 2022 ] PDG discovered [33] 246919023530449916313 which is smaller than Feng Yuan's record.
On [ July 31, 2022 ] PDG discovered [34] 332369949023486562727 which is smaller than Feng Yuan's record.
So it seems Feng Yuan didn't search exhaustively !

[ November 9, 2022 ]
David Griffeath (email) took over and confirmed voids for 44, 45, 46 and 47-digit palindromic octagonals,
and shared his 48-digit palindromic octagonal record [38].

[ December 6, 2022 ]
Robert Xiao (email) added four new entries [39] up to [42].

[ January 8, 2023 ]
David Griffeath's (email) laptop cranked out an exhaustive search of 53-digit sporadic octagonals.
[43] is his unique find, a new record.

[ June 15 & 16, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [44] up to [46].

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