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Numbers whose XOR-triangles have reflection symmetry.
1

%I #16 May 25 2020 04:21:26

%S 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,

%T 62,63,64,65,72,73,85,86,93,94,96,99,104,107,118,119,126,127,128,129,

%U 153,158,165,168,182,189,195,200,214,219,224,231,254,255,256,257

%N Numbers whose XOR-triangles have reflection symmetry.

%C There are three possible axes of symmetry:

%C .

%C . V

%C . U W

%C . ._____._____.

%C . \ . . /

%C . \ . /

%C . . .

%C . . \ . / .

%C . W \ / U

%C . .

%C .

%C . V

%C .

%C - symmetry through axis U-U is only possible for the numbers 0 and 1,

%C - symmetry through axis V-V corresponds to binary palindromes,

%C - symmetry through axis W-W corresponds to number k such that A334727(k) is a binary palindrome,

%C - 0 and 1 are the only terms whose XOR-triangles have the three symmetries,

%C - XOR-triangles of other terms have only one kind of symmetry.

%H Rémy Sigrist, <a href="/A334918/b334918.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A334918/a334918.png">Triangles illustrating the initial terms</a> (binary palindromes are rendered in red)

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>

%e The XOR-triangles for a(15) = 21 and a(16) = 22 are as follows (with dots instead of 0's for clarity):

%e 1 . 1 . 1 1 . 1 1 .

%e 1 1 1 1 1 1 . 1

%e . . . . 1 1

%e . . 1 .

%e . 1

%o (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)) }

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

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, May 16 2020