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Palindromic Octagonals (or 8-gonals)
Factorization Records triangle square penta hexa hepta nona

  left to right (forwards) as from the right to left (backwards)

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 )

So far this compilation counts 46 Palindromic Octagonals.

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.

There exist no palindromic octagonals of length

A B C →  3 4 1 

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

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

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

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

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

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

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

• From Eric Weisstein's Math Encyclopedia : Octagonal Number

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

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

[ January 3, 2008 ]
Feng Yuan (email) from Washington State, USA, discovered the palindromic octagonals
with index numbers [32] and [36].

Exhaustive search performed up to length 43 by Patrick De Geest and found [37] on [ July 17, 2022 ].

On [ July 5, 2022 ] PDG discovered [35] 736230589620506244343 which is smaller than Feng Yuan's record.
On [ July 30, 2022 ] PDG discovered [33] 246919023530449916313 which is smaller than Feng Yuan's record.
On [ July 31, 2022 ] PDG discovered [34] 332369949023486562727 which is smaller than Feng Yuan's record.
So it seems Feng Yuan didn't search exhaustively !

[ November 9, 2022 ]
David Griffeath (email) took over and confirmed voids for 44, 45, 46 and 47-digit palindromic octagonals,
and shared his 48-digit palindromic octagonal record [38].

[ December 6, 2022 ]
Robert Xiao (email) added four new entries [39] up to [42].

[ January 8, 2023 ]
David Griffeath's (email) laptop cranked out an exhaustive search of 53-digit sporadic octagonals.
[43] is his unique find, a new record.

[ June 15 & 16, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [44] up to [46].

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( © All rights reserved ) - Last modified : July 3, 2023.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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