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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 :

 PLAIN TEXT POLYGONALS

Normal and Palindromic Pentagonals

So far I compiled 59 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 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, 37, 38, 39, 42, 45, 46.
(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 by Patrick De Geest.
Higher lengths up to length 47 were found and submitted by Feng Yuan.

Index NrInfo BasenumberLength
Palindromic PentagonalsLength

Formula = n(3n–1)/2
59Info128.481.111.918.761.748.263.90824
24.761.094.179.822.073.249.932.323.994.237.022.897.149.016.74247
58Info4.181.240.246.129.576.678.50322
26.224.154.993.780.584.443.322.334.448.508.739.945.142.26244
57Info1.161.338.314.703.013.743.74322
2.023.060.021.795.854.282.372.732.824.585.971.200.603.20243
56Info101.908.013.398.232.552.38121
15.577.864.792.161.518.113.031.181.516.129.746.877.55141
55Info32.178.927.183.122.554.46120
1.553.225.031.985.055.486.446.845.505.891.305.223.55140
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
17InfoPrime 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

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

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

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

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