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Palindromic Heptagonals
(or 7-gonals)
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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

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

base x ( 5 x base - 3 )
-------------------------------
2
The best way to get a 'structural' insight as how to imagine heptagonals is to visit for instance this site :


Normal and Palindromic Heptagonals

flash So far 64 Palindromic Heptagonals are compiled.

The largest one was discovered by Feng Yuan from Washington State, USA,
on [ January 6, 2008 ].

This basenumber
115.095.375.604.446.234.345.993
has 24 digits
yielding the following palindromic heptagonal number
33.117.363.713.821.393.460.337.673.306.439.312.831.736.371.133
with a length of 47 digits.

bu17 A palindromic heptagonal numbers can end with any digit.

bu17 There exist no palindromic heptagonals of length  8, 9, 14, 16, 32.

(Sloane's A059869)

bu17 Sloane's A046194 gives the first numbers that are both Heptagonal and Triangular.

1, 55, 121771, 5720653, 12625478965, ...
Consult also Eric Weisstein's page Heptagonal Triangular Number.

bu17 Sloane's A036354 gives the first numbers that are both Heptagonal and Square.

1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, ...
Consult also Eric Weisstein's page Heptagonal Square Number.

bu17 Sloane's A048900 gives the first numbers that are both Heptagonal and Pentagonal.

1, 4347, 16701685, 246532939589097, ...
Consult also Eric Weisstein's page Heptagonal Pentagonal Number.

bu17 Sloane's A048903 gives the first numbers that are both Heptagonal and Hexagonal.

1, 121771, 12625478965, 1309034909945503, ...
Consult also Eric Weisstein's page Heptagonal Hexagonal Number.

bu17 Sloane's A048906 gives the first numbers that are both Heptagonal and Octagonal.

1, 297045, 69010153345, ...
Consult also Eric Weisstein's page Heptagonal Octagonal Number.

bu17 Sloane's A048921 gives the first numbers that are both Heptagonal and Nonagonal.

1, 26884, 542041975, 10928650279834, ...
Consult also Eric Weisstein's page Heptagonal Nonagonal Number.


Sources Revealed


Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
One can find the regular heptagonal numbers at
%N Heptagonal (or 7-gonal) numbers n(5n–3)/2 under A000566.
The palindromic heptagonal numbers are categorised as follows :
%N n(5n–3)/2 is a palindromic heptagonal number under A054971.
%N Palindromic heptagonal numbers under A054910.
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 NrInfo BasenumberLength
Palindromic HeptagonalsLength
   
Formula = n(5n–3)/2
64 Info 115.095.375.604.446.234.345.99324
33.117.363.713.821.393.460.337.673.306.439.312.831.736.371.13347
63 Info 1.645.934.853.472.802.341.77622
6.772.753.854.691.338.287.602.067.828.331.964.583.572.77643
62 Info 1.197.770.348.245.060.425.83822
3.586.634.517.837.733.317.833.387.133.377.387.154.366.85343
61 Info 584.961.928.628.774.900.04321
855.451.144.862.739.850.023.320.058.937.268.441.154.55842
60 Info 149.974.969.674.243.257.99021
56.231.228.821.975.462.213.831.226.457.912.882.213.26541
59 Info 26.678.135.232.269.748.22120
1.779.307.248.678.181.132.112.311.818.768.427.039.77140
58 Info Prime Curios!          22.059.819.805.104.914.32120
1.216.589.124.584.247.550.110.557.424.854.219.856.12140
57 Info 10.317.264.079.491.259.30720
266.114.845.214.901.555.595.555.109.412.548.411.66239
56 Info 1.743.257.048.371.139.57719
7.597.362.841.739.144.174.714.419.371.482.637.95737
55 Info 386.569.698.677.663.35818
373.590.329.839.348.608.806.843.938.923.095.37336
54 Info 140.620.708.714.288.53918
49.435.459.298.271.961.416.917.289.295.453.49435
53 Info 25.901.555.713.068.48617
1.677.226.470.892.976.776.792.980.746.227.76134
52 Info 8.570.921.185.931.80616
183.651.724.938.636.676.636.839.427.156.38133
51 Info 1.019.248.876.540.67216
2.597.170.680.823.553.553.280.860.717.95231
50 Info 627.391.088.906.93415
984.048.946.099.569.965.990.649.840.48930
49 Info 523.047.977.062.61115
683.947.965.773.223.322.377.569.749.38630
48 Info 186.276.561.018.50315
86.747.392.962.199.899.126.929.374.76829
47 Info 139.689.569.097.50415
48.782.939.286.615.651.668.293.928.78429
46 Info 87.036.116.082.28114
18.938.213.756.720.602.765.731.283.98129
45 Info 47.892.275.461.25014
5.734.175.122.140.550.412.215.714.37528
44 Info 43.909.485.680.87914
4.820.107.331.898.228.981.337.010.28428
43 Info 10.024.819.305.87214
251.242.505.288.444.882.505.242.15227
42 Info 6.639.848.560.86113
110.218.972.277.909.772.279.812.01127
41 Info 3.129.887.444.61213
24.490.488.539.844.893.588.409.44226
40 Info 1.755.516.804.29713
7.704.598.125.420.245.218.954.07725
39 Info 1.648.402.664.97613
6.793.078.364.747.474.638.703.97625
38 Info 987.706.147.38712
2.438.908.583.963.693.858.098.34225
37 Info 498.373.323.65112
620.939.924.316.613.429.939.02624
36 Info 178.357.794.54212
79.528.757.184.448.175.782.59723
35 Info 159.052.425.33612
63.244.185.012.921.058.144.23623
34 Info 147.826.795.94512
54.631.903.998.189.930.913.64523
33 Info Prime Curios!          71.784.986.52111
12.882.710.724.442.701.728.82123
32 Info 36.987.715.85311
3.420.227.810.000.187.220.24322
31 Info 29.970.725.86711
2.245.611.022.442.201.165.42222
30 Info 17.663.032.82211
779.956.821.151.128.659.97721
29 Info 15.911.875.33611
632.969.441.747.144.969.23621
28 Info 2.850.685.77210
20.316.023.422.432.061.30220
27 Info 2.497.949.42610
15.599.378.333.387.399.55120
26 Info 1.820.198.06310
8.282.802.468.642.082.828
ps : all digits are even !		19
25 Info 765.609.0419
1.465.393.008.003.935.64119
24 Info 496.329.4169
615.857.222.222.758.51618
23 Info 88.274.1418
19.480.809.790.808.49117
22 Info 16.943.2628
717.685.292.586.71715
21 Info 1.686.5377
7.111.015.101.11713
20 Info 1.207.6987
3.646.334.336.46313
19 Info 615.7696
947.927.729.74912
18 Info 382.1736
365.139.931.56312
17 Info 371.7786
345.546.645.54312
16 Info 190.9146
91.120.102.11911
15 Info 171.6576
73.665.056.63711
14 Info 161.6566
65.331.413.35611
13 Info 158.4766
62.786.368.72611
12 Info 53.5375
7.165.445.61710
11 Info 1.5564
6.050.5067
10 Info 1.4704
5.400.0457
9 Info 1.3594
4.615.1647
8 Info 5833
848.8486
7 Info 1563
60.6065
6 Info 442
4.7744
5 Info 382
3.5534
4 Info 162
6163
3 Info Prime!          51
552
2 Info Prime!          21
Prime!          71
1 Info 11
11


Contributions

Feng Yuan (email) from Washington State, USA, discovered the palindromic heptagonals
starting from index number 55 up to 64 [ January 8, 2008 ].

Feng Yuan (email from Washington State, USA, discovered the palindromic heptagonals
starting from index number 53 up to 54.

Jeff Heleen (email submitted the first 52 palindromic heptagonals.

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(All rights reserved) - Last modified : June 1, 2009.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com