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Palindromic Pentagonals
Factorization Records triangle square hexa hepta octa nona

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

Pentagonal 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 - 1 ) ------------------------------- 2

So far I compiled 76 Palindromic Pentagonals.

All palindromic pentagonal numbers can only end with 0, 1, 2, 5, 6 or 7.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.

There exist no palindromic pentagonals of length  3, 9, 11, 12, 24, 30, 32, 33, 42, 45, 46, 50, 51, 52, 54.

A B C →  6 5 1 

A B C →  6 11 5 

Believe it or not but some numbers can't be expressed as the sum of 3 pentagonal numbers.

Every pentagonal number is one-third of a triangular number.

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'.

Sloane's A014979 gives the first numbers that are both Pentagonal and Triangular.

Sloane's A036353 gives the first numbers that are both Pentagonal and Square.

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

Sloane's A048900 gives the first numbers that are both Pentagonal and Heptagonal.

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

Sloane's A048915 gives the first numbers that are both Pentagonal and Nonagonal.

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

• From Eric Weisstein's Math Encyclopedia : Pentagonal Number

Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences One can find the regular pentagonal numbers at %N Pentagonal numbers n(3n–1)/2 under A000326. The palindromic pentagonal numbers are categorised as follows : %N n(3n–1)/2 is a pentagonal palindrome under A028386. %N Pentagonal palindromes under A002069. 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.

Exhaustive search performed up to length 29 by Patrick De Geest.

Higher lengths up to length 47 were found and submitted by Feng Yuan.

Warut Roonguthai (email) from Thailand helped searching for these pentagonal palindromes.
He discovered those starting from index number [26] up to [36].

Feng Yuan (email) from Washington State, USA, discovered the palindromic pentagonals
starting from index number [50] and [51].

[ January 8, 2008 ]
Feng Yuan (email) from Washington State, USA, submitted the palindromic pentagonals
starting from index number [52] up to [59].

[ November 4, 2022 ]
Robert Xiao (email) added three missing entries [55], [56] and [57].

[ November 11, 2022 ]
Robert Xiao (email) added two missing entries [60] and [62].

[ November 11, 2022 ]
David Griffeath (email) using Rust code by Robert Xiao (email) added two entries [65] and [66].

[ December 6, 2022 ]
Robert Xiao (email) added three new record entries [67], [68] and [69].

[ January 9, 2023 ]
David Griffeath's (email) laptop cranked out an exhaustive search of 53-digit sporadic pentagonals
just beyond the cutoff of Robert's global search up to length 52. In total six new entries [70] up to [75] were added.

[ June 16, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added one more entry [76].

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( © All rights reserved ) - Last modified : July 2, 2023.

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

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