A045654 Number of 2n-bead balanced binary strings, rotationally equivalent to complement.

1, 2, 6, 8, 22, 32, 72, 128, 278, 512, 1056, 2048, 4168, 8192, 16512, 32768, 65814, 131072, 262656, 524288, 1049632, 2097152, 4196352, 8388608, 16781384, 33554432, 67117056, 134217728, 268451968, 536870912, 1073774592, 2147483648

Offset

0,2

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Formula

a(0)=1, a(2n) = a(n)+2^(2n), a(2n+1) = 2^(2n+1). - Ralf Stephan, Jun 07 2003

G.f.: 1/(1-x) + sum(k>=0, t(1+2t-2t^2)/(1-t^2)/(1-2t), t=x^2^k). - Ralf Stephan, Aug 30 2003

For n >= 1, a(n) = sum(0<=k<=A007814(n), 2^(n/2^k)). - David W. Wilson, Jan 01 2012

Inverse Moebius transform of A045663. - Andrew Howroyd, Sep 15 2019

Prog

(PARI) a(n)={if(n==0, 1, my(s=0); while(n%2==0, s+=2^n; n/=2); s + 2^n)} \\ Andrew Howroyd, Sep 22 2019
(Python)
def A045654(n): return sum(1<<(n>>k) for k in range((~n & n-1).bit_length()+1)) if n else 1 # Chai Wah Wu, Jul 22 2024

Crossrefs

Cf. A000984, A045653, A045655, A045656, A045663.

Sequence in context: A045656 A343256 A129342 * A081672 A332348 A137072

Adjacent sequences: A045651 A045652 A045653 * A045655 A045656 A045657

Keyword

nonn

Author

David W. Wilson

Status

approved