A334918 Numbers whose XOR-triangles have reflection symmetry.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 21, 22, 24, 27, 30, 31, 32, 33, 40, 45, 51, 54, 62, 63, 64, 65, 72, 73, 85, 86, 93, 94, 96, 99, 104, 107, 118, 119, 126, 127, 128, 129, 153, 158, 165, 168, 182, 189, 195, 200, 214, 219, 224, 231, 254, 255, 256, 257

Offset: 1

Author

Rémy Sigrist, May 16 2020

Keywords

• nonn
• base

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.

Examples

			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
		

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)
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to XOR-triangles

Crossrefs

Cf. A006995, A334556 (rotational symmetry), A334727.

Programs

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)) }