OFFSET
0,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
Tilman Piesk, Row sums of triangle derived from A385665
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf 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_{k=0..A007814(n)} 2^(n/2^k). - David W. Wilson, Jan 01 2012
Inverse Moebius transform of A045663. - Andrew Howroyd, Sep 15 2019
a(n) = 2*A127804(n-1) for n > 0. - Tilman Piesk, Jul 05 2025
a(n) = Sum_{k=1..n} 2 * n * A385665(n,k) / k. - Tilman Piesk, Jul 07 2025
EXAMPLE
From Andrew Howroyd, Jul 06 2025: (Start)
The a(1) = 2 length 2 balanced binary strings are: 01, 10.
The a(2) = 6 strings are: 0101, 1010, 0011, 0110, 1100, 1001.
The a(3) = 8 strings are: 010101, 101010, 000111, 001110, 011100, 111000, 110001, 100011. (End)
MAPLE
a:= proc(n) option remember;
2^n+`if`(n::even and n>0, a(n/2), 0)
end:
seq(a(n), n=0..33); # Alois P. Heinz, Jul 01 2025
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved