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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 :

     PLAIN TEXT POLYGONS 

Normal and Palindromic Heptagonals
flash So far 65 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.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
2 can be followed by any number : 20, 21, 22, 23, 24, 25, 26, 27, 28 or 29
3 can be followed by any number : 30, 31, 32, 33, 34, 35, 36, 37, 38 or 39
4 can be followed by any number : 40, 41, 42, 43, 44, 45, 46, 47, 48 or 49
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
6 can be followed by any number : 60, 61, 62, 63, 64, 65, 66, 67, 68 or 69
7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79
8 can be followed by any number : 80, 81, 82, 83, 84, 85, 86, 87, 88 or 89
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
bu17 There exist no palindromic heptagonals of length  8, 9, 14, 16, 32, 38, 44, 45, 46.
(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
65 Info115.095.375.604.446.234.345.99324
33.117.363.713.821.393.460.337.673.306.439.312.831.736.371.13347
64 Info1.645.934.853.472.802.341.77622
6.772.753.854.691.338.287.602.067.828.331.964.583.572.77643
63 Info1.197.770.348.245.060.425.83822
3.586.634.517.837.733.317.833.387.133.377.387.154.366.85343
62 Info584.961.928.628.774.900.04321
855.451.144.862.739.850.023.320.058.937.268.441.154.55842
61 Info149.974.969.674.243.257.99021
56.231.228.821.975.462.213.831.226.457.912.882.213.26541
60 Info26.678.135.232.269.748.22120
1.779.307.248.678.181.132.112.311.818.768.427.039.77140
59 InfoPrime Curios!    22.059.819.805.104.914.32120
1.216.589.124.584.247.550.110.557.424.854.219.856.12140
58 Info10.317.264.079.491.259.30720
266.114.845.214.901.555.595.555.109.412.548.411.66239
57 Info1.743.257.048.371.139.57719
7.597.362.841.739.144.174.714.419.371.482.637.95737
56 Info386.569.698.677.663.35818
373.590.329.839.348.608.806.843.938.923.095.37336
55 Info140.620.708.714.288.53918
49.435.459.298.271.961.416.917.289.295.453.49435
54 Info25.901.555.713.068.48617
1.677.226.470.892.976.776.792.980.746.227.76134
53 Info8.570.921.185.931.80616
183.651.724.938.636.676.636.839.427.156.38133
52 Info1.019.248.876.540.67216
2.597.170.680.823.553.553.280.860.717.95231
51 Info627.391.088.906.93415
984.048.946.099.569.965.990.649.840.48930
50 Info523.047.977.062.61115
683.947.965.773.223.322.377.569.749.38630
49 Info186.276.561.018.50315
86.747.392.962.199.899.126.929.374.76829
48 Info139.689.569.097.50415
48.782.939.286.615.651.668.293.928.78429
47 Info87.036.116.082.28114
18.938.213.756.720.602.765.731.283.98129
46 Info47.892.275.461.25014
5.734.175.122.140.550.412.215.714.37528
45 Info43.909.485.680.87914
4.820.107.331.898.228.981.337.010.28428
44 Info10.024.819.305.87214
251.242.505.288.444.882.505.242.15227
43 Info6.639.848.560.86113
110.218.972.277.909.772.279.812.01127
42 Info3.129.887.444.61213
24.490.488.539.844.893.588.409.44226
41 Info1.755.516.804.29713
7.704.598.125.420.245.218.954.07725
40 Info1.648.402.664.97613
6.793.078.364.747.474.638.703.97625
39 Info987.706.147.38712
2.438.908.583.963.693.858.098.34225
38 Info498.373.323.65112
620.939.924.316.613.429.939.02624
37 Info178.357.794.54212
79.528.757.184.448.175.782.59723
36 Info159.052.425.33612
63.244.185.012.921.058.144.23623
35 Info147.826.795.94512
54.631.903.998.189.930.913.64523
34 InfoPrime Curios!    71.784.986.52111
12.882.710.724.442.701.728.82123
33 Info36.987.715.85311
3.420.227.810.000.187.220.24322
32 Info29.970.725.86711
2.245.611.022.442.201.165.42222
31 Info17.663.032.82211
779.956.821.151.128.659.97721
30 Info15.911.875.33611
632.969.441.747.144.969.23621
29 Info2.850.685.77210
20.316.023.422.432.061.30220
28 Info2.497.949.42610
15.599.378.333.387.399.55120
27 Info1.820.198.06310
Only even digits !    8.282.802.468.642.082.82819
26 Info765.609.0419
1.465.393.008.003.935.64119
25 Info496.329.4169
615.857.222.222.758.51618
24 Info88.274.1418
19.480.809.790.808.49117
23 Info16.943.2628
717.685.292.586.71715
22 Info1.686.5377
7.111.015.101.11713
21 Info1.207.6987
3.646.334.336.46313
20 Info615.7696
947.927.729.74912
19 Info382.1736
365.139.931.56312
18 Info371.7786
345.546.645.54312
17 Info190.9146
91.120.102.11911
16 Info171.6576
73.665.056.63711
15 Info161.6566
65.331.413.35611
14 Info158.4766
62.786.368.72611
13 Info53.5375
7.165.445.61710
12 Info1.5564
6.050.5067
11 Info1.4704
5.400.0457
10 Info1.3594
4.615.1647
9 Info5833
848.8486
8 Info1563
60.6065
7 Info442
4.7744
6 Info382
3.5534
5 Info162
6163
4 InfoPrime!    51
552
3 InfoPrime!    21
Prime!    71
2 Info11
11
1 Info01
01


Contributions

Feng Yuan (email) from Washington State, USA, discovered the palindromic heptagonals
starting from index number 56 up to 65 [ January 8, 2008 ].

Feng Yuan (email from Washington State, USA, discovered the palindromic heptagonals
starting from index number 54 up to 55.

Jeff Heleen (email submitted the first 53 palindromic heptagonals.






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( © All rights reserved ) - Last modified : September 2, 2021.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com