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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

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

base x ( 2 x base - 1 )

 PLAIN TEXT POLYGONS

Normal and Palindromic Hexagonals

So far this compilation counts 83 Palindromic Hexagonals.
Here is the largest Palindromic Hexagonal that Patrick De Geest
discovered, using CUDA code by Robert Xiao, on [ September 18, 2023 ]

 This basenumber5.802.177.948.883.358.568.632.104.907.376 has 31 digits yielding the following palindromic hexagonal number 67.330.537.901.016.595.837.936.819.195.144.159.191.863.973.859.561.010.973.503.376 with a length of 62 digits.

Hexagonal numbers, whether palindromic or not, can only end with the following digits 0, 1, 3, 5, 6 or 8.
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 : 35 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 exist no palindromic hexagonals of length  3, 8, 9, 12, 22, 24, 27, 30, 36, 38, 40, 42, 43, 46, 47, 48, 50, 52, 56, 57, 60, 61.
(Sloane's A082721)

 Case Hexa Change of variables CUDApalin parameters Base Correction base n = m + 1 `A B C →  2 3 1 ` base = CUDAbase + 1

Every hexagonal number is a triangular number.

A good source for such statements is “The Book of Numbers”
by John H. Conway and Richard K. Guy.
Click on the image on the left for more background about the book.
The book can be ordered at 'www.amazon.com'.

The set of hexagonal numbers is a subset of the triangular numbers.
In fact every other triangular number Tn is a hexagonal number,
with Hn = T2n–1. So in practice this means that whenever you have
an odd basenumber of a triangular (it doesn't matter if it is palindromic
or not) add 1 and divide by 2 and you'll have your basenumber of
a hexagonal number.
For instance BaseT = 3185 gives (3185+1)/2 or BaseH = 1593.
Both basenumbers yield resp. 5073705 ! (See T20 and H13 in the list
of palindromic triangulars and hexagonals.)

Sloane's A046177 gives the first numbers that are both Hexagonal and Square.
1, 1225, 1413721, 1631432881, 1882672131025, ...
Consult also Eric Weisstein's page Hexagonal Square Number.

Sloane's A046180 gives the first numbers that are both Hexagonal and Pentagonal.
1, 40755, 1533776805, 57722156241751, ...
Consult also Eric Weisstein's page Hexagonal Pentagonal Number.

Sloane's A048903 gives the first numbers that are both Hexagonal and Heptagonal.
1, 121771, 12625478965, 1309034909945503, ...
Consult also Eric Weisstein's page Hexagonal Heptagonal Number.

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

Sloane's A048918 gives the first numbers that are both Hexagonal and Nonagonal.
1, 325, 5330229625, 1353857339341, 22184715227362706161, ...
Consult also Eric Weisstein's page Hexagonal Nonagonal Number.

The best way to get a 'structural' insight as how to imagine hexagonals 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 hexagonal numbers at %N Hexagonal numbers n(2n–1) under A000384. The palindromic hexagonal numbers are categorised as follows : %N n(2n–1) is a hexagonal palindrome under A054970. %N Hexagonal palindromes under A054969. 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 Hexagonal NumberLength

[PG6] Formula = n(2n–1)
83
T195
Info5.802.177.948.883.358.568.632.104.907.37631
67.330.537.901.016.595.837.936.819.195.144.159.191.863.973.859.561.010.973.503.37662
82
T191
Info2.978.027.651.105.307.283.560.165.296.24731
17.737.297.381.495.587.611.197.659.145.355.354.195.679.111.678.559.418.379.273.77162
81
T189
Info2.755.156.575.179.618.913.249.404.977.38731
15.181.775.507.510.974.169.398.336.861.133.116.863.389.396.147.901.570.557.718.15162
80
T183
Info166.343.895.877.755.240.864.992.331.74330
55.340.583.391.578.954.029.492.500.160.806.100.529.492.045.987.519.338.504.35559
79
T181
Info30.930.709.782.292.023.763.863.665.30729
1.913.417.615.272.750.984.206.227.878.228.787.226.024.890.572.725.167.143.19158
78
T179
Info16.486.144.312.213.259.790.973.168.70529
543.585.908.566.243.233.447.813.416.222.614.318.744.332.342.665.809.585.34558
77
T177
Info9.616.108.721.198.505.559.428.274.76128
184.939.093.875.819.915.846.819.675.717.576.918.648.519.918.578.390.939.48158
76
T176
Info2.085.761.317.162.842.560.211.961.59428
8.700.800.544.345.751.829.458.446.650.566.448.549.281.575.434.450.080.07855
75
T173
Info847.467.228.857.802.971.885.663.75727
1.436.401.407.975.847.596.488.073.464.643.708.846.957.485.797.041.046.34155
74
T172
Info804.421.800.530.291.612.930.873.24127
1.294.188.866.336.792.535.757.443.701.073.447.575.352.976.336.688.814.92155
73
T170
Info263.852.272.444.989.543.277.818.91127
139.236.043.348.769.976.436.410.774.477.014.634.679.967.843.340.632.93154
72
T169
Info232.381.414.676.970.907.940.093.60127
108.002.243.774.540.620.212.510.333.333.015.212.026.045.477.342.200.80154
71
T168
Info158.794.471.171.971.963.309.802.29327
50.431.368.149.572.469.821.594.569.096.549.512.896.427.594.186.313.40553
70
T167
Info97.498.711.546.492.941.662.293.00726
19.011.997.506.452.472.127.996.132.423.169.972.127.425.460.579.911.09153
69
T166
Info96.669.112.362.709.388.798.156.77726
18.689.834.569.988.266.379.838.452.825.483.897.366.288.996.543.898.68153
68
T161
Info17.285.022.733.311.846.957.809.18326
597.544.021.782.214.705.597.988.959.889.795.507.412.287.120.445.79551
67
T160
Info8.056.799.672.501.327.442.134.69725
129.824.041.925.634.994.253.924.353.429.352.499.436.529.140.428.92151
66
T156
Info1.640.970.415.884.137.024.922.82325
5.385.567.811.613.915.254.381.275.721.834.525.193.161.187.655.83549
65
T154
Info986.044.423.140.796.944.147.20724
1.944.567.208.814.134.024.246.830.386.424.204.314.188.027.654.49149
64
T151
Info712.999.646.411.249.822.712.23724
1.016.736.991.565.134.544.383.803.083.834.454.315.651.996.376.10149
63
T147
Info13.269.591.374.387.166.732.74923
352.164.110.486.420.593.101.030.101.395.024.684.011.461.25345
62
T146
Info12.432.083.271.361.292.035.37923
309.113.388.932.122.569.558.171.855.965.221.239.883.311.90345
61
T144
Info2.902.825.573.527.004.575.02122
16.852.792.620.644.766.088.388.388.066.744.602.629.725.86144
60
T138
Info166.735.172.845.051.317.04321
55.601.235.727.338.276.212.021.267.283.372.753.210.65541
59
T137
Info95.800.231.113.659.236.06120
18.355.368.562.861.046.305.450.364.016.826.586.355.38141
58
T136
Info17.684.912.039.911.051.42620
625.512.227.718.781.732.353.237.187.817.722.215.52639
57
T134
Info Prime Curios!    9.561.677.372.927.686.36119
182.851.348.367.914.603.505.306.419.763.843.158.28139
56
T131
Info1.827.172.728.272.717.28219
6.677.120.357.887.130.286.820.317.887.530.217.76637
55
T127
Info1.666.614.952.453.588.94319
5.555.210.799.483.757.064.607.573.849.970.125.55537
54
T122
Info967.877.687.832.504.86118
1.873.574.437.207.991.455.541.997.027.344.753.78137
53
T118
Info123.592.172.047.762.89918
30.550.049.982.967.649.594.676.928.994.005.50335
52
T117
Info56.270.393.370.816.59217
6.332.714.340.212.879.669.782.120.434.172.33634
51
T116
Info54.556.973.614.741.45517
5.952.926.739.999.190.550.919.999.376.292.59534
50
T115
Info54.050.101.714.919.96517
5.842.826.990.786.388.228.836.870.996.282.48534
49
T114
Info39.179.467.697.750.41917
3.070.061.378.158.136.996.318.518.731.600.70334
48
T113
Info13.277.954.159.263.74917
352.608.133.311.018.969.810.113.331.806.25333
47
T111
Info2.250.495.266.170.70116
10.129.457.886.113.466.431.168.875.492.10132
46
T109
Info1.864.517.945.162.97216
6.952.854.335.669.501.059.665.334.582.59631
45
T108
Info1.624.954.860.315.51316
5.280.956.596.126.015.106.216.956.590.82531
44
T106
Info176.760.310.346.46215
62.488.414.627.554.945.572.641.488.42629
43
T105
Info163.412.656.430.01515
53.407.392.563.028.082.036.529.370.43529
42
T103
Info82.201.095.219.11114
13.514.040.110.442.624.401.104.041.53129
41
T101
Info25.182.304.403.14114
1.268.296.910.104.884.010.196.928.62128
40
T97
Info5.711.952.918.93613
65.252.812.296.277.269.221.825.25626
39
T93
Info2.031.643.008.98413
8.255.146.631.905.091.366.415.52825
38
T89
Info208.949.080.21412
87.319.436.244.344.263.491.37823
37
T88
Info202.635.166.86412
82.122.021.699.799.612.022.12823
36
T86
Info176.234.896.62612
62.117.477.577.377.577.471.12623
35
T79
Info Prime!    13.347.462.82911
356.309.527.929.725.903.65321
34
T76
Info8.763.870.45110
153.610.850.555.058.016.35121
33
T73
Info2.756.800.38710
15.199.896.744.769.899.15120
32
T72
Info1.783.816.19610
6.364.000.440.440.004.63619
31
T70
Info1.240.058.21910
3.075.488.771.778.845.70319
30
T69
Info507.354.4039
514.816.979.979.618.41518
29
T67
Info184.252.8769
67.898.244.444.289.87617
28
T66
Info181.737.1829
66.056.806.460.865.06617
27
T65
Info170.563.6739
58.183.932.923.938.18517
26
T64
Info91.071.9218
16.588.189.498.188.56117
25
T57
Info25.004.4418
1.250.444.114.440.52116
24
T52
Info17.835.1968
636.188.414.881.63615
23
T51
Info17.352.5868
602.224.464.422.20615
22
T42
Info5.555.5567
61.728.399.382.71614
21
T40
Info1.853.9427
6.874.200.024.78613
20
T38
Info1.620.2137
5.250.178.710.52513
19
T37
Info810.3116
1.313.207.023.13113
18
T30
Info83.5275
13.953.435.93111
17
T29
Info55.5565
6.172.882.71610
16
T28
Info51.4255
5.289.009.82510
15
T26
Info25.1415
1.264.114.62110
14
T21
Info1.6854
5.676.7657
13
T20
Info1.5934
5.073.7057
12
T18
Info9173
1.680.8617
11
T17
InfoPrime!    7973
1.269.6217
10
T16
Info6443
828.8286
9
T15
Info5563
617.7166
8
T14
Info1823
66.0665
7
T13
Info872
15.0515
6
T11
Info552
5.9954
5
T10
Info392
3.0034
4
T6
Info61
662
3
T3
InfoPrime!    21
61
2
T2
Info11
11
1
T1
Info01
01

Contributions

Feng Yuan (email) from Washington State, USA, helped making the list of palindromic hexagonals
starting from index number [1] up to [52].

[ January 8, 2008 ]
Feng Yuan (email) from Washington State, USA, sent in the palindromic hexagonals
with index number [61] and [63].

[ November 6, 2022 ]
David Griffeath (email) using Rust code from Robert Xiao added three more record entries [64], [65] and [66]
and two more [67] and [68] on [ November 9, 2022 ].

[ November 21, 2022 ]
David Griffeath (email) using CUDA code from Robert Xiao added five more record entries [69] up to [73].

[ December 6, 2022 ]
Robert Xiao (email) added one entry missed by Feng Yuan. See index nbr [62].

[ June 14, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [74] up to [76].

[ June 26, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [77] up to [79].

[ September 6, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added one more entry [80].

[ September 24, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [81] up to [83].

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