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Palindromic Octagonals
(or 8-gonals)
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Introduction
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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

Octagonal 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 - 2 )
The best way to get a 'structural' insight as how to imagine octagonals is to visit for instance this site :

Normal and Palindromic Octagonals
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flash So far I compiled 32 Palindromic Octagonals.

The largest one was found by Feng Yuan from Washington State, USA,
on [ January 3, 2008 ].

This basenumber
790.350.914.396.051.463.771
has 21 digits
yielding the following palindromic octagonal number
1.873.963.703.660.024.006.763.676.004.200.663.073.693.781
with a length of 43 digits.

bu17 A palindromic octagonal numbers can only end with digit 1, 3, 5, 6 and 8.

bu17 There exist no palindromic octagonals of length

2, 3, 5, 6, 7, 8, 10, 11, 15, 17, 18, 20, 22, 23, 26, 31, 34.
(Sloane's A059870)



Sources Revealed
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Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
One can find the regular octagonal numbers at
%N Octagonal numbers: n(3n–2) under A000567.
The palindromic octagonal numbers are categorised as follows :
%N n(3n–2) is a palindromic octagonal number under A057106.
%N Palindromic octagonal numbers under A057107.
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
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Exhaustive search performed upto  length 37

Index NrInfo BasenumberLength
Palindromic OctagonalsLength
   
Formula = n(3n–2)
32 Info 790.350.914.396.051.463.77121
1.873.963.703.660.024.006.763.676.004.200.663.073.693.78143
31 Info 65.621.182.548.683.007.15320
12.918.418.797.262.737.739.193.773.726.279.781.481.92141
30 Info 704.687.633.763.271.76118
1.489.753.983.536.637.090.907.366.353.893.579.84137
29 Info 589.588.144.333.920.23318
1.042.842.539.817.346.669.666.437.189.352.482.40137
28 Info 221.420.450.951.946.11118
147.081.048.299.289.519.915.982.992.840.180.74136
27 Info 145.815.635.485.061.31818
63.786.598.655.736.820.002.863.755.689.568.73635
26 Info 136.626.014.502.187.50518
56.000.003.516.255.851.015.855.261.530.000.06535
25 Info 7.897.699.388.697.56316
187.120.966.902.701.565.107.209.669.021.78133
24 Info 4.731.413.168.091.25816
67.158.811.701.562.055.026.510.711.885.17632
23 Info 222.032.718.762.72315
147.895.584.603.498.894.306.485.598.74130
22 Info 105.802.560.494.38715
33.582.545.421.505.050.512.454.528.53329
21 Info 78.849.864.240.62114
18.651.903.272.292.929.227.230.915.68129
20 Info 51.722.791.547.84214
8.025.741.496.504.444.056.941.475.20828
19 Info 5.847.307.263.80113
102.573.006.711.888.117.600.375.20127
18 Info 1.400.295.262.09513
5.882.480.463.134.313.640.842.88525
17 Info 335.038.979.07712
336.753.352.502.205.253.357.63324
16 Info 13.175.129.92511
520.752.145.595.541.257.02521
15 Info 1.050.553.50710
3.310.988.011.108.890.13319
14 Info 42.687.0158
5.466.543.663.456.64516
13 Info 2.017.6317
12.212.500.521.22114
12 Info 1.472.7467
6.506.939.396.05613
11 Info 1.054.1677
3.333.802.083.33313
10 Info 1.054.0677
3.333.169.613.33313
9 Info 453.4866
616.947.749.61612
8 Info 426.1156
544.721.127.44512
7 Info 7.8634
185.464.5819
6 Info 7.3414
161.656.1619
5 Info 6.4314
124.060.4219
4 Info 6.3314
120.232.0219
3 Info 522
8.0084
2 Info Prime!          21
81
1 Info 11
11

bu17 Sloane's A046183 gives the first numbers that are both Octagonal and Triangular.

1, 21, 11781, 203841, 113123361, ...
Consult also Eric Weisstein's page Octagonal Triangular Number.

bu17 Sloane's A036428 gives the first numbers that are both Octagonal and Square.

1, 225, 43681, 8473921, 1643897025, ...
Consult also Eric Weisstein's page Octagonal Square Number.

bu17 Sloane's A046189 gives the first numbers that are both Octagonal and Pentagonal.

1, 176, 1575425, 234631320, 2098015778145, ...
Consult also Eric Weisstein's page Octagonal Pentagonal Number.

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

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

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

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

bu17 Sloane's A048924 gives the first numbers that are both Octagonal and Nonagonal.

1, 631125, 286703855361, 130242107189808901, ...
Consult also Eric Weisstein's page Octagonal Nonagonal Number.



Contributions
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Feng Yuan from Washington State, USA, discovered the palindromic octagonals
with index numbers 31 & 32 [ January 3, 2008 ].

Feng Yuan from Washington State, USA, discovered the palindromic octagonals
starting from index number 17 up to 30.

The Lowell Family submitted the first 7 palindromic octagonals.

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(All rights reserved) - Last modified : January 17, 2008.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com