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Palindromic Octagonals
(or 8-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

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 :

     PLAIN TEXT POLYGONS 

Normal and Palindromic Octagonals

flash So far I compiled 33 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 0, 1, 3, 5, 6 and 8.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can only be followed by an even number : 10, 12, 14, 16 or 18
3 can only be followed by 3 : 33
5 can only be followed by an even number : 50, 52, 54, 56 or 58
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
8 can only be followed by 0 : 80

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, 38, 39, 40, 42.
(Sloane's A059870)


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.


Sources Revealed


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

Exhaustive search performed upto length 37 by Patrick De Geest
Higher values up to length 43 were found and submitted by Feng Yuan.



Index NrInfo BasenumberLength
Palindromic OctagonalsLength
   
Formula = n(3n–2)
33 Info790.350.914.396.051.463.77121
1.873.963.703.660.024.006.763.676.004.200.663.073.693.78143
32 Info65.621.182.548.683.007.15320
12.918.418.797.262.737.739.193.773.726.279.781.481.92141
31 Info704.687.633.763.271.76118
1.489.753.983.536.637.090.907.366.353.893.579.84137
30 Info589.588.144.333.920.23318
1.042.842.539.817.346.669.666.437.189.352.482.40137
29 Info221.420.450.951.946.11118
147.081.048.299.289.519.915.982.992.840.180.74136
28 Info145.815.635.485.061.31818
63.786.598.655.736.820.002.863.755.689.568.73635
27 Info136.626.014.502.187.50518
56.000.003.516.255.851.015.855.261.530.000.06535
26 Info7.897.699.388.697.56316
187.120.966.902.701.565.107.209.669.021.78133
25 Info4.731.413.168.091.25816
67.158.811.701.562.055.026.510.711.885.17632
24 Info222.032.718.762.72315
147.895.584.603.498.894.306.485.598.74130
23 Info105.802.560.494.38715
33.582.545.421.505.050.512.454.528.53329
22 Info78.849.864.240.62114
18.651.903.272.292.929.227.230.915.68129
21 Info51.722.791.547.84214
8.025.741.496.504.444.056.941.475.20828
20 Info5.847.307.263.80113
102.573.006.711.888.117.600.375.20127
19 Info1.400.295.262.09513
5.882.480.463.134.313.640.842.88525
18 Info335.038.979.07712
336.753.352.502.205.253.357.63324
17 Info13.175.129.92511
520.752.145.595.541.257.02521
16 Info1.050.553.50710
3.310.988.011.108.890.13319
15 Info42.687.0158
5.466.543.663.456.64516
14 Info2.017.6317
12.212.500.521.22114
13 Info1.472.7467
6.506.939.396.05613
12 Info1.054.1677
3.333.802.083.33313
11 Info1.054.0677
3.333.169.613.33313
10 Info453.4866
616.947.749.61612
9 Info426.1156
544.721.127.44512
8 Info7.8634
185.464.5819
7 Info7.3414
161.656.1619
6 Info6.4314
124.060.4219
5 Info6.3314
120.232.0219
4 Info522
8.0084
3 InfoPrime!    21
81
2 Info11
11
1 Info01
01


Contributions

Feng Yuan (email) from Washington State, USA, discovered the palindromic octagonals
with index numbers 32 & 33 [ January 3, 2008 ].

Feng Yuan (email) from Washington State, USA, discovered the palindromic octagonals
starting from index number 18 up to 31.

The Lowell Family (email) submitted the first 8 palindromic octagonals.





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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