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Palindromic Hexagonals
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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

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

base x ( 2 x base - 1 )
The best way to get a 'structural' insight as how to imagine hexagonals is to visit for instance this site :


Normal and Palindromic Hexagonals

flash So far this compilation counts 61 Palindromic Hexagonals.

Here is the largest one that Feng Yuan discovered on [ January 4, 2008 ]

This basenumber
13.269.591.374.387.166.732.749
has 23 digits
yielding the following palindromic hexagonal number
352.164.110.486.420.593.101.030.101.395.024.684.011.461.253
with a length of 45 digits.

bu17 All palindromic hexagonal numbers can only end with 1, 3, 5, 6 or 8.

bu17 There exist no palindromic hexagonals of length  3, 8, 9, 12, 22, 24, 27, 30, 36, 38, 40.

(Sloane's A082721)

bu17 Every hexagonal number is 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 A046177 gives the first numbers that are both Hexagonal and Square.

1, 1225, 1413721, 1631432881, 1882672131025, ...
Consult also Eric Weisstein's page Hexagonal Square Number.

bu17 Sloane's A046180 gives the first numbers that are both Hexagonal and Pentagonal.

1, 40755, 1533776805, 57722156241751, ...
Consult also Eric Weisstein's page Hexagonal Pentagonal Number.

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

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

bu17 Sloane's A046192 gives the first numbers that are both Hexagonal and Octagonal.

1, 11781, 113123361, 1086210502741, ...
Consult also Eric Weisstein's page Hexagonal Octagonal Number.

bu17 Sloane's A048918 gives the first numbers that are both Hexagonal and Nonagonal.

1, 325, 5330229625, 1353857339341, 22184715227362706161, ...
Consult also Eric Weisstein's page Hexagonal Nonagonal Number.


Sources Revealed


Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
One can find the regular hexagonal numbers at
%N Hexagonal numbers n(2n–1) under A000384.
The palindromic hexagonal numbers are categorised as follows :
%N n(2n–1) is a hexagonal palindrome under A054970.
%N Hexagonal palindromes under A054969.
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 Nr Info BasenumberLength
Palindromic Hexagonal NumberLength
   
Formula = n(2n–1)
61
T144		 Info 13.269.591.374.387.166.732.74923
352.164.110.486.420.593.101.030.101.395.024.684.011.461.25345
60
T143		 Info 2.902.825.573.527.004.575.02122
16.852.792.620.644.766.088.388.388.066.744.602.629.725.86144
59
T137		 Info 166.735.172.845.051.317.04321
55.601.235.727.338.276.212.021.267.283.372.753.210.65541
58
T136		 Info 95.800.231.113.659.236.06120
18.355.368.562.861.046.305.450.364.016.826.586.355.38141
57
T135		 Info 17.684.912.039.911.051.42620
625.512.227.718.781.732.353.237.187.817.722.215.52639
56
T133		 Info Prime Curios!          9.561.677.372.927.686.36119
182.851.348.367.914.603.505.306.419.763.843.158.28139
55
T130		 Info 1.827.172.728.272.717.28219
6.677.120.357.887.130.286.820.317.887.530.217.76637
54
T126		 Info 1.666.614.952.453.588.94319
5.555.210.799.483.757.064.607.573.849.970.125.55537
53
T121		 Info 967.877.687.832.504.86118
1.873.574.437.207.991.455.541.997.027.344.753.78137
52
T117		 Info 123.592.172.047.762.89918
30.550.049.982.967.649.594.676.928.994.005.50335
51
T116		 Info 56.270.393.370.816.59217
6.332.714.340.212.879.669.782.120.434.172.33634
50
T115		 Info 54.556.973.614.741.45517
5.952.926.739.999.190.550.919.999.376.292.59534
49
T114		 Info 54.050.101.714.919.96517
5.842.826.990.786.388.228.836.870.996.282.48534
48
T113		 Info 39.179.467.697.750.41917
3.070.061.378.158.136.996.318.518.731.600.70334
47
T112		 Info 13.277.954.159.263.74917
352.608.133.311.018.969.810.113.331.806.25333
46
T110		 Info 2.250.495.266.170.70116
10.129.457.886.113.466.431.168.875.492.10132
45
T108		 Info 1.864.517.945.162.97216
6.952.854.335.669.501.059.665.334.582.59631
44
T107		 Info 1.624.954.860.315.51316
5.280.956.596.126.015.106.216.956.590.82531
43
T105		 Info 176.760.310.346.46215
62.488.414.627.554.945.572.641.488.42629
42
T104		 Info 163.412.656.430.01515
53.407.392.563.028.082.036.529.370.43529
41
T102		 Info 82.201.095.219.11114
13.514.040.110.442.624.401.104.041.53129
40
T100		 Info 25.182.304.403.14114
1.268.296.910.104.884.010.196.928.62128
39
T96		 Info 5.711.952.918.93613
65.252.812.296.277.269.221.825.25626
38
T92		 Info 2.031.643.008.98413
8.255.146.631.905.091.366.415.52825
37
T88		 Info 208.949.080.21412
87.319.436.244.344.263.491.37823
36
T87		 Info 202.635.166.86412
82.122.021.699.799.612.022.12823
35
T85		 Info 176.234.896.62612
62.117.477.577.377.577.471.12623
34
T78		 Info Prime!          13.347.462.82911
356.309.527.929.725.903.65321
33
T75		 Info 8.763.870.45110
153.610.850.555.058.016.35121
32
T72		 Info 2.756.800.38710
15.199.896.744.769.899.15120
31
T71		 Info 1.783.816.19610
6.364.000.440.440.004.63619
30
T69		 Info 1.240.058.21910
3.075.488.771.778.845.70319
29
T68		 Info 507.354.4039
514.816.979.979.618.41518
28
T66		 Info 184.252.8769
67.898.244.444.289.87617
27
T65		 Info 181.737.1829
66.056.806.460.865.06617
26
T64		 Info 170.563.6739
58.183.932.923.938.18517
25
T63		 Info 91.071.9218
16.588.189.498.188.56117
24
T56		 Info 25.004.4418
1.250.444.114.440.52116
23
T51		 Info 17.835.1968
636.188.414.881.63615
22
T50		 Info 17.352.5868
602.224.464.422.20615
21
T41		 Info 5.555.5567
61.728.399.382.71614
20
T39		 Info 1.853.9427
6.874.200.024.78613
19
T37		 Info 1.620.2137
5.250.178.710.52513
18
T36		 Info 810.3116
1.313.207.023.13113
17
T29		 Info 83.5275
13.953.435.93111
16
T28		 Info 55.5565
6.172.882.71610
15
T27		 Info 51.4255
5.289.009.82510
14
T25		 Info 25.1415
1.264.114.62110
13
T20		 Info 1.6854
5.676.7657
12
T19		 Info 1.5934
5.073.7057
11
T17		 Info 9173
1.680.8617
10
T16		 Info Prime!          7973
1.269.6217
9
T15		 Info 6443
828.8286
8
T14		 Info 5563
617.7166
7
T13		 Info 1823
66.0665
6
T12		 Info 872
15.0515
5
T10		 Info 552
5.9954
4
T9		 Info 392
3.0034
3
T5		 Info 61
662
2
T3		 Info Prime!          21
61
1
T1		 Info 11
11


Contributions

Feng Yuan (email) from Washington State, USA, sent in the palindromic hexagonals
with index number 60 and 61 on [ January 8, 2008 ].

Feng Yuan (email) from Washington State, USA, helped making the list of palindromic hexagonals
starting from index number 1 up to 51.

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(All rights reserved) - Last modified : November 23, 2011.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com