A138954 Number of complement symmetries in the rotations of the binary expansion of a number.

0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,11

Comments

It seems that the number of complement rotational symmetries is nonzero iff #0 = #1 in the binary expansion of a number.

The above statement is true in only one direction. It is clearly necessary for the number of 1 bits to equal the number of 0 bits. However, this is not sufficient. The first counterexample is n = 37 with binary expansion 100101 and complement 011010. Values of n for which a(n) is nonzero are therefore a proper subset of A031443. - Andrew Howroyd, Jan 12 2020

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1023

Example

a(2) = 1 because 2 has binary expansion 10 and the complement shows up once in rotations;
a(10) = 2 because 10 has binary expansion 1010 and its complement shows up twice in rotations.

Prog

(PARI) a(n)={my(s=0); if(n, my(b=logint(n, 2)+1); if(2*hammingweight(n)==b, my(w=2^b-1-n); for(i=2, b, w=if(w%2, w+2^b, w)\2; if(w==n, s++)))); s} \\ Andrew Howroyd, Jan 12 2020

Crossrefs

Cf. A031443, A138904.

Sequence in context: A350022 A348264 A127282 * A245811 A258822 A353373

Adjacent sequences: A138951 A138952 A138953 * A138955 A138956 A138957

Keyword

easy,nonn

Author

Max Sills, Apr 03 2008

Extensions

Missing a(8) inserted and terms a(21) and beyond from Andrew Howroyd, Jan 12 2020

Status

approved