OFFSET
1,3
COMMENTS
There are three possible axes of symmetry:
.
. V
. U W
. ._____._____.
. \ . . /
. \ . /
. . .
. . \ . / .
. W \ / U
. .
.
. V
.
- symmetry through axis U-U is only possible for the numbers 0 and 1,
- symmetry through axis V-V corresponds to binary palindromes,
- symmetry through axis W-W corresponds to number k such that A334727(k) is a binary palindrome,
- 0 and 1 are the only terms whose XOR-triangles have the three symmetries,
- XOR-triangles of other terms have only one kind of symmetry.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Triangles illustrating the initial terms (binary palindromes are rendered in red)
EXAMPLE
The XOR-triangles for a(15) = 21 and a(16) = 22 are as follows (with dots instead of 0's for clarity):
1 . 1 . 1 1 . 1 1 .
1 1 1 1 1 1 . 1
. . . . 1 1
. . 1 .
. 1
PROG
(PARI) is(n) = { my (b=binary(n)); if (b==Vecrev(b), return (1), my (w=#b-1, x=n); for (k=0, w, if (bittest(n, k)!=bittest(x, 0), return (0)); x=bitxor(x, x\2)); return (1)) }
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, May 16 2020
STATUS
approved