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Palindromic Hexagonals
Factorization Records triangle square penta hepta octa nona

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

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 )

So far this compilation counts 83 Palindromic Hexagonals.

Hexagonal numbers, whether palindromic or not, can only end with the following digits 0, 1, 3, 5, 6 or 8.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.

There exist no palindromic hexagonals of length  3, 8, 9, 12, 22, 24, 27, 30, 36, 38, 40, 42, 43, 46, 47, 48, 50, 52, 56, 57, 60, 61.

A B C →  2 3 1 

Every hexagonal number is 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'.

The set of hexagonal numbers is a subset of the triangular numbers.
In fact every other triangular number T_n is a hexagonal number,
with H_n = T_2n–1. So in practice this means that whenever you have
an odd basenumber of a triangular (it doesn't matter if it is palindromic
or not) add 1 and divide by 2 and you'll have your basenumber of
a hexagonal number.
For instance Base_T = 3185 gives (3185+1)/2 or Base_H = 1593.
Both basenumbers yield resp. 5073705 ! (See T₂₀ and H₁₃ in the list
of palindromic triangulars and hexagonals.)

Sloane's A046177 gives the first numbers that are both Hexagonal and Square.

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

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

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

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

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

• From Eric Weisstein's Math Encyclopedia : Hexagonal Number

Feng Yuan (email) from Washington State, USA, helped making the list of palindromic hexagonals
starting from index number [1] up to [52].

[ January 8, 2008 ]
Feng Yuan (email) from Washington State, USA, sent in the palindromic hexagonals
with index number [61] and [63].

[ November 6, 2022 ]
David Griffeath (email) using Rust code from Robert Xiao added three more record entries [64], [65] and [66]
and two more [67] and [68] on [ November 9, 2022 ].

[ November 21, 2022 ]
David Griffeath (email) using CUDA code from Robert Xiao added five more record entries [69] up to [73].

[ December 6, 2022 ]
Robert Xiao (email) added one entry missed by Feng Yuan. See index nbr [62].

[ June 14, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [74] up to [76].

[ June 26, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [77] up to [79].

[ September 6, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added one more entry [80].

[ September 24, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [81] up to [83].

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