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

Normal and Palindromic Pentagonals

So far I compiled 58 Palindromic Pentagonals.
Here is the largest one that Feng Yuan from Washington State, USA,
discovered on [ January 6, 2008 ].

 This basenumber128.481.111.918.761.748.263.908 has 24 digits yielding the following palindromic pentagonal number 24.761.094.179.822.073.249.932.323.994.237.022.897.149.016.742 with a length of 47 digits.

All palindromic pentagonal numbers can only end with 1, 2, 5, 6 or 7.

There exist no palindromic pentagonals of length  3, 9, 11, 12, 24.
(Sloane's A059868)

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.

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

Exhaustive search performed upto length 29

Index NrInfo BasenumberLength
Palindromic PentagonalsLength

Formula = n(3n–1)/2
58 Info 128.481.111.918.761.748.263.90824
24.761.094.179.822.073.249.932.323.994.237.022.897.149.016.74247
57 Info 4.181.240.246.129.576.678.50322
26.224.154.993.780.584.443.322.334.448.508.739.945.142.26244
56 Info 1.161.338.314.703.013.743.74322
2.023.060.021.795.854.282.372.732.824.585.971.200.603.20243
55 Info 101.908.013.398.232.552.38121
15.577.864.792.161.518.113.031.181.516.129.746.877.55141
54 Info 32.178.927.183.122.554.46120
1.553.225.031.985.055.486.446.845.505.891.305.223.55140
53 Info 642.912.991.153.879.99618
620.005.671.291.643.466.664.346.192.176.500.02636
52 Info 222.497.695.855.916.65418
74.257.836.991.787.986.368.978.719.963.875.24735
51 Info 64.541.848.291.142.55617
6.248.475.271.255.291.881.925.521.725.748.42634
50 Info 1.343.985.615.421.96316
2.709.446.001.691.728.271.961.006.449.07231
49 Info 1.331.901.641.608.90316
2.660.942.974.380.735.370.834.792.490.66231
48 Info 192.460.705.375.19015
55.561.684.670.273.437.207.648.616.55529
47 Info 189.271.045.290.60515
53.735.292.878.097.279.087.829.253.73529
46 Info 62.640.018.209.36214
5.885.657.821.903.773.091.287.565.88528
45 Info 62.407.875.352.93714
5.842.114.359.101.551.019.534.112.48528
44 Info 22.073.207.482.88914
730.839.732.873.989.378.237.938.03727
43 Info 8.505.462.902.90613
108.514.348.789.060.987.843.415.80127
42 Info 6.106.091.136.27713
55.926.523.446.777.764.432.562.95526
41 Info 3.690.877.000.94313
20.433.859.554.133.145.595.833.40226
40 Info 2.704.211.220.54613
10.969.137.487.988.978.473.196.90126
39 Info 2.269.269.065.47413
7.724.373.137.274.727.313.734.27725
38 Info 2.118.668.671.85113
6.733.135.411.623.261.145.313.37625
37 Info 1.199.186.171.66413
2.157.071.211.464.641.121.707.51225
36 Info 100.208.264.58112
15.062.544.435.453.444.526.05123
35 Info 72.507.025.78911
7.885.903.183.113.813.095.88722
34 Info 20.483.169.15611
629.340.327.999.723.043.92621
33 Info 18.517.422.79711
514.342.420.555.024.243.41521
32 Info 18.306.458.26511
502.689.621.303.126.986.20521
31 Info 6.693.818.69610
67.210.813.099.031.801.27620
30 Info 2.049.123.15610
6.298.358.561.658.538.92619
29 Info 1.345.679.68810
2.716.280.733.370.826.17219
28 Info 851.649.3069
1.087.959.810.189.597.80119
27 Info 818.049.2019
1.003.806.742.476.083.00119
26 Info 700.457.1539
735.960.334.433.069.53718
25 Info 316.882.0869
150.621.384.483.126.05118
24 Info 225.563.5339
76.318.361.016.381.36717
23 Info 100.058.5819
15.017.579.397.571.05117
22 Info 84.885.7618
10.808.388.588.380.80117
21 Info 59.401.6508
5.292.834.004.382.92516
20 Info 13.280.8398
264.571.020.175.46215
19 Info 2.684.5617
10.810.300.301.80114
18 Info 1.979.1307
5.875.432.345.78513
17 Info 1.965.1177
5.792.526.252.97513
16 Info Prime Curios!          1.258.7237
2.376.574.756.73213
15 Info Prime!          44.0595
2.911.771.19210
14 Info 31.9265
1.528.888.25110
13 Info 26.9065
1.085.885.80110
12 Info 26.4665
1.050.660.50110
11 Info 6.0104
54.177.1458
10 Info 4.2284
26.811.8628
9 Info 2.2294
7.451.5477
8 Info 2.1734
7.081.8077
7 Info 6933
720.0276
6 Info Prime!          1013
15.2515
5 Info 442
2.8824
4 Info 262
1.0014
3 Info 41
222
2 Info Prime!          21
51
1 Info 11
11

Contributions

Feng Yuan (email) from Washington State, USA, submitted the palindromic pentagonals
starting from index number 51 up to 58 on [ January 8, 2008 ].

Feng Yuan (email) from Washington State, USA, discovered the palindromic pentagonals
starting from index number 49 up to 50.

Warut Roonguthai (email) from Thailand helped searching for these pentagonal palindromes.
He discovered those starting from index number 25 up to 35.

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