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

1,3

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)

Index entries for sequences related to binary expansion of n

Index entries for sequences related to XOR-triangles

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

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

Sequence in context: A174876 A092597 A335134 * A354220 A125506 A079334

Adjacent sequences: A334915 A334916 A334917 * A334919 A334920 A334921

Keyword

nonn,base

Author

Rémy Sigrist, May 16 2020

Status

approved