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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

Pentagonal 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 - 1 )
-------------------------------
2

 PLAIN TEXT POLYGONALS

Normal and Palindromic Pentagonals

So far I compiled 76 Palindromic Pentagonals.
Here is the largest Sporadic Palindromic Pentagonal that Patrick De Geest
discovered, using CUDA code by Robert Xiao, on [ June 16, 2023 ]

 This basenumber829.412.616.490.882.316.063.874.786 has 27 digits yielding the following palindromic pentagonal number 1.031.887.932.591.377.142.105.479.799.979.745.012.417.731.952.397.881.301 with a length of 55 digits.

All palindromic pentagonal numbers can only end with 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 0 or 5 : 10 or 15
2 can be followed by any number : 20, 21, 22, 23, 24, 25, 26, 27, 28 or 29
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
6 can only be followed by 2 or 7 : 62 or 67
7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79

There exist no palindromic pentagonals of length  3, 9, 11, 12, 24, 30, 32, 33, 42, 45, 46, 50, 51, 52, 54.
(Sloane's A059868)

 Case Penta Change of variables CUDApalin parameters Base Correction odd base n = 2 * m + 1 `A B C →  6 5 1 ` CUDAbase * 2 + 1 even base n = 2 * m + 2 `A B C →  6 11 5 ` CUDAbase * 2 + 2

Believe it or not but some numbers can't be expressed as the sum of 3 pentagonal numbers.
%N Not the sum of 3 pentagonal numbers. under Sloane's A003679.
4, 8, 9, 16, 19, ...

Every pentagonal number is one-third of 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'.

Sloane's A014979 gives the first numbers that are both Pentagonal and Triangular.
(Sequence corrected and extended by Warut Roonguthai.)
1, 210, 40755, 7906276, 1533776805, 297544793910, ...
Consult also Eric Weisstein's page Pentagonal Triangular Number.

Sloane's A036353 gives the first numbers that are both Pentagonal and Square.
1, 9801, 94109401, 903638458801, 8676736387298001, ...
Consult also Eric Weisstein's page Pentagonal Square Number.

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

Sloane's A048900 gives the first numbers that are both Pentagonal and Heptagonal.
1, 4347, 16701685, 246532939589097, ...
Consult also Eric Weisstein's page Pentagonal Heptagonal Number.

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

Sloane's A048915 gives the first numbers that are both Pentagonal and Nonagonal.
1, 651, 180868051, 95317119801, 26472137730696901, ...
Consult also Eric Weisstein's page Pentagonal Nonagonal Number.

The best way to get a 'structural' insight as how to imagine pentagonals 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 pentagonal numbers at %N Pentagonal numbers n(3n–1)/2 under A000326. The palindromic pentagonal numbers are categorised as follows : %N n(3n–1)/2 is a pentagonal palindrome under A028386. %N Pentagonal palindromes under A002069. 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 PentagonalsLength

[PG5] Formula = n(3n–1)/2
76Info829.412.616.490.882.316.063.874.78627
1.031.887.932.591.377.142.105.479.799.979.745.012.417.731.952.397.881.30155
75Info219.838.383.896.945.839.241.529.09327
72.493.372.551.631.403.519.074.864.446.847.091.530.413.615.527.339.42753
74Info211.567.096.617.431.127.327.768.33127
67.140.954.556.694.156.338.058.353.235.385.083.365.149.665.545.904.17653
73Info196.744.195.862.034.221.724.924.90527
58.062.417.908.097.725.661.820.084.648.002.816.652.779.080.971.426.08553
72Info140.815.521.353.961.270.534.020.20827
29.743.516.581.281.883.860.142.882.928.824.106.838.818.218.561.534.79253
71Info123.470.453.704.578.756.803.923.20427
22.867.429.407.021.789.074.918.527.272.581.947.098.712.070.492.476.82253
70Info115.716.935.699.622.852.551.776.54327
20.085.613.811.565.974.693.953.734.743.735.939.647.956.511.831.658.00253
69Info1.997.352.210.935.459.325.350.51025
5.984.123.781.793.151.403.874.475.744.783.041.513.971.873.214.89549
68Info1.366.335.687.983.609.354.348.84425
2.800.309.818.386.464.643.692.792.972.963.464.646.838.189.030.08249
67Info1.214.693.643.268.411.793.685.38825
2.213.220.970.495.031.472.095.671.765.902.741.305.940.790.223.12249
66Info646.108.029.840.746.975.319.15124
626.183.379.337.037.375.893.787.787.398.573.730.733.973.381.62648
65Info386.808.141.157.899.128.841.06324
224.430.807.099.043.827.014.991.199.410.728.340.990.708.034.42248
64Info128.481.111.918.761.748.263.90824
24.761.094.179.822.073.249.932.323.994.237.022.897.149.016.74247
63Info4.181.240.246.129.576.678.50322
26.224.154.993.780.584.443.322.334.448.508.739.945.142.26244
62Info3.166.575.018.238.865.276.80622
15.040.796.019.201.704.940.688.604.940.710.291.069.704.05144
61Info1.161.338.314.703.013.743.74322
2.023.060.021.795.854.282.372.732.824.585.971.200.603.20243
60Info226.255.113.072.008.674.10921
76.787.064.286.841.145.600.200.654.114.868.246.078.76741
59Info101.908.013.398.232.552.38121
15.577.864.792.161.518.113.031.181.516.129.746.877.55141
58Info32.178.927.183.122.554.46120
1.553.225.031.985.055.486.446.845.505.891.305.223.55140
57Info12.470.286.104.989.637.21920
233.262.053.310.446.425.999.524.644.013.350.262.33239
56Info3.225.069.471.604.675.44619
15.601.609.645.014.690.722.709.641.054.690.610.65138
55Info2.215.732.656.045.463.37819
7.364.206.804.599.425.576.755.249.954.086.024.63737
54Info642.912.991.153.879.99618
620.005.671.291.643.466.664.346.192.176.500.02636
53Info222.497.695.855.916.65418
74.257.836.991.787.986.368.978.719.963.875.24735
52Info64.541.848.291.142.55617
6.248.475.271.255.291.881.925.521.725.748.42634
51Info1.343.985.615.421.96316
2.709.446.001.691.728.271.961.006.449.07231
50Info1.331.901.641.608.90316
2.660.942.974.380.735.370.834.792.490.66231
49Info192.460.705.375.19015
55.561.684.670.273.437.207.648.616.55529
48Info189.271.045.290.60515
53.735.292.878.097.279.087.829.253.73529
47Info62.640.018.209.36214
5.885.657.821.903.773.091.287.565.88528
46Info62.407.875.352.93714
5.842.114.359.101.551.019.534.112.48528
45Info22.073.207.482.88914
730.839.732.873.989.378.237.938.03727
44Info8.505.462.902.90613
108.514.348.789.060.987.843.415.80127
43Info6.106.091.136.27713
55.926.523.446.777.764.432.562.95526
42Info3.690.877.000.94313
20.433.859.554.133.145.595.833.40226
41Info2.704.211.220.54613
10.969.137.487.988.978.473.196.90126
40Info2.269.269.065.47413
7.724.373.137.274.727.313.734.27725
39Info2.118.668.671.85113
6.733.135.411.623.261.145.313.37625
38Info1.199.186.171.66413
2.157.071.211.464.641.121.707.51225
37Info100.208.264.58112
15.062.544.435.453.444.526.05123
36Info72.507.025.78911
7.885.903.183.113.813.095.88722
35Info20.483.169.15611
629.340.327.999.723.043.92621
34Info18.517.422.79711
514.342.420.555.024.243.41521
33Info18.306.458.26511
502.689.621.303.126.986.20521
32Info6.693.818.69610
67.210.813.099.031.801.27620
31Info2.049.123.15610
6.298.358.561.658.538.92619
30Info1.345.679.68810
2.716.280.733.370.826.17219
29Info851.649.3069
1.087.959.810.189.597.80119
28Info818.049.2019
1.003.806.742.476.083.00119
27Info700.457.1539
735.960.334.433.069.53718
26Info316.882.0869
150.621.384.483.126.05118
25Info225.563.5339
76.318.361.016.381.36717
24Info100.058.5819
15.017.579.397.571.05117
23Info84.885.7618
10.808.388.588.380.80117
22Info59.401.6508
5.292.834.004.382.92516
21Info13.280.8398
264.571.020.175.46215
20Info2.684.5617
10.810.300.301.80114
19Info1.979.1307
5.875.432.345.78513
18Info1.965.1177
5.792.526.252.97513
17Info Prime Curios!    1.258.7237
2.376.574.756.73213
16InfoPrime!    44.0595
2.911.771.19210
15Info31.9265
1.528.888.25110
14Info26.9065
1.085.885.80110
13Info26.4665
1.050.660.50110
12Info6.0104
54.177.1458
11Info4.2284
26.811.8628
10Info2.2294
7.451.5477
9Info2.1734
7.081.8077
8Info6933
720.0276
7InfoPrime!    1013
15.2515
6Info442
2.8824
5Info262
1.0014
4Info41
222
3InfoPrime!    21
Prime!    51
2Info11
11
1Info01
01

Contributions

Exhaustive search performed up to length 29 by Patrick De Geest.

Higher lengths up to length 47 were found and submitted by Feng Yuan.

Warut Roonguthai (email) from Thailand helped searching for these pentagonal palindromes.
He discovered those starting from index number [26] up to [36].

Feng Yuan (email) from Washington State, USA, discovered the palindromic pentagonals
starting from index number [50] and [51].

[ January 8, 2008 ]
Feng Yuan (email) from Washington State, USA, submitted the palindromic pentagonals
starting from index number [52] up to [59].

[ November 4, 2022 ]
Robert Xiao (email) added three missing entries [55], [56] and [57].

[ November 11, 2022 ]
Robert Xiao (email) added two missing entries [60] and [62].

[ November 11, 2022 ]
David Griffeath (email) using Rust code by Robert Xiao (email) added two entries [65] and [66].

[ December 6, 2022 ]
Robert Xiao (email) added three new record entries [67], [68] and [69].

[ January 9, 2023 ]
David Griffeath's (email) laptop cranked out an exhaustive search of 53-digit sporadic pentagonals
just beyond the cutoff of Robert's global search up to length 52. In total six new entries [70] up to [75] were added.

[ June 16, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added one more entry [76].

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