World!Of
Numbers
HOME plate
WON |

EMBED
Palindromic Pentagonals pentagon
rood Factorization rood Records rood
rood triangle rood square rood hexa rood hepta rood octa rood nona



Introduction

Palindromic numbers are numbers which read the same from
 p_right left to right (forwards) as from the right to left (backwards) p_left
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

flash 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 basenumber
128.481.111.918.761.748.263.908
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.


bu17 All palindromic pentagonal numbers can only end with 1, 2, 5, 6 or 7.

bu17 There exist no palindromic pentagonals of length  3, 9, 11, 12, 24.
(Sloane's A059868)


bu17 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, ...

bu17 Every pentagonal number is one-third of a triangular number.
book
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'.


bu17 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.

bu17 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.

bu17 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.

bu17 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.

bu17 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.

bu17 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
58 Info128.481.111.918.761.748.263.90824
24.761.094.179.822.073.249.932.323.994.237.022.897.149.016.74247
57 Info4.181.240.246.129.576.678.50322
26.224.154.993.780.584.443.322.334.448.508.739.945.142.26244
56 Info1.161.338.314.703.013.743.74322
2.023.060.021.795.854.282.372.732.824.585.971.200.603.20243
55 Info101.908.013.398.232.552.38121
15.577.864.792.161.518.113.031.181.516.129.746.877.55141
54 Info32.178.927.183.122.554.46120
1.553.225.031.985.055.486.446.845.505.891.305.223.55140
53 Info642.912.991.153.879.99618
620.005.671.291.643.466.664.346.192.176.500.02636
52 Info222.497.695.855.916.65418
74.257.836.991.787.986.368.978.719.963.875.24735
51 Info64.541.848.291.142.55617
6.248.475.271.255.291.881.925.521.725.748.42634
50 Info1.343.985.615.421.96316
2.709.446.001.691.728.271.961.006.449.07231
49 Info1.331.901.641.608.90316
2.660.942.974.380.735.370.834.792.490.66231
48 Info192.460.705.375.19015
55.561.684.670.273.437.207.648.616.55529
47 Info189.271.045.290.60515
53.735.292.878.097.279.087.829.253.73529
46 Info62.640.018.209.36214
5.885.657.821.903.773.091.287.565.88528
45 Info62.407.875.352.93714
5.842.114.359.101.551.019.534.112.48528
44 Info22.073.207.482.88914
730.839.732.873.989.378.237.938.03727
43 Info8.505.462.902.90613
108.514.348.789.060.987.843.415.80127
42 Info6.106.091.136.27713
55.926.523.446.777.764.432.562.95526
41 Info3.690.877.000.94313
20.433.859.554.133.145.595.833.40226
40 Info2.704.211.220.54613
10.969.137.487.988.978.473.196.90126
39 Info2.269.269.065.47413
7.724.373.137.274.727.313.734.27725
38 Info2.118.668.671.85113
6.733.135.411.623.261.145.313.37625
37 Info1.199.186.171.66413
2.157.071.211.464.641.121.707.51225
36 Info100.208.264.58112
15.062.544.435.453.444.526.05123
35 Info72.507.025.78911
7.885.903.183.113.813.095.88722
34 Info20.483.169.15611
629.340.327.999.723.043.92621
33 Info18.517.422.79711
514.342.420.555.024.243.41521
32 Info18.306.458.26511
502.689.621.303.126.986.20521
31 Info6.693.818.69610
67.210.813.099.031.801.27620
30 Info2.049.123.15610
6.298.358.561.658.538.92619
29 Info1.345.679.68810
2.716.280.733.370.826.17219
28 Info851.649.3069
1.087.959.810.189.597.80119
27 Info818.049.2019
1.003.806.742.476.083.00119
26 Info700.457.1539
735.960.334.433.069.53718
25 Info316.882.0869
150.621.384.483.126.05118
24 Info225.563.5339
76.318.361.016.381.36717
23 Info100.058.5819
15.017.579.397.571.05117
22 Info84.885.7618
10.808.388.588.380.80117
21 Info59.401.6508
5.292.834.004.382.92516
20 Info13.280.8398
264.571.020.175.46215
19 Info2.684.5617
10.810.300.301.80114
18 Info1.979.1307
5.875.432.345.78513
17 Info1.965.1177
5.792.526.252.97513
16 InfoPrime Curios!    1.258.7237
2.376.574.756.73213
15 InfoPrime!    44.0595
2.911.771.19210
14 Info31.9265
1.528.888.25110
13 Info26.9065
1.085.885.80110
12 Info26.4665
1.050.660.50110
11 Info6.0104
54.177.1458
10 Info4.2284
26.811.8628
9 Info2.2294
7.451.5477
8 Info2.1734
7.081.8077
7 Info6933
720.0276
6 InfoPrime!    1013
15.2515
5 Info442
2.8824
4 Info262
1.0014
3 Info41
222
2 InfoPrime!    21
51
1 Info11
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.






[up TOP OF PAGE]

comments


( © All rights reserved ) - Last modified : January 28, 2021.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com