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A334918 Numbers whose XOR-triangles have reflection symmetry.

%I A334918 #16 May 25 2020 04:21:26
%S A334918 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 A334918 62,63,64,65,72,73,85,86,93,94,96,99,104,107,118,119,126,127,128,129,
%U A334918 153,158,165,168,182,189,195,200,214,219,224,231,254,255,256,257
%N A334918 Numbers whose XOR-triangles have reflection symmetry.
%C A334918 There are three possible axes of symmetry:
%C A334918 .
%C A334918 .                V
%C A334918 .       U                 W
%C A334918 .          ._____._____.
%C A334918 .           \ .     . /
%C A334918 .            \   .   /
%C A334918 .             .     .
%C A334918 .          .   \ . /   .
%C A334918 .       W       \ /       U
%C A334918 .                .
%C A334918 .
%C A334918 .                V
%C A334918 .
%C A334918 - symmetry through axis U-U is only possible for the numbers 0 and 1,
%C A334918 - symmetry through axis V-V corresponds to binary palindromes,
%C A334918 - symmetry through axis W-W corresponds to number k such that A334727(k) is a binary palindrome,
%C A334918 - 0 and 1 are the only terms whose XOR-triangles have the three symmetries,
%C A334918 - XOR-triangles of other terms have only one kind of symmetry.
%H A334918 Rémy Sigrist, <a href="/A334918/b334918.txt">Table of n, a(n) for n = 1..10000</a>
%H A334918 Rémy Sigrist, <a href="/A334918/a334918.png">Triangles illustrating the initial terms</a> (binary palindromes are rendered in red)
%H A334918 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A334918 <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>
%e A334918 The XOR-triangles for a(15) = 21 and a(16) = 22 are as follows (with dots instead of 0's for clarity):
%e A334918                       1 . 1 . 1      1 . 1 1 .
%e A334918                        1 1 1 1        1 1 . 1
%e A334918                         . . .          . 1 1
%e A334918                          . .            1 .
%e A334918                           .              1
%o A334918 (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 A334918 Cf. A006995, A334556 (rotational symmetry), A334727.
%K A334918 nonn,base
%O A334918 1,3
%A A334918 _Rémy Sigrist_, May 16 2020