A101691 A modular binomial sum sequence.

1, 1, 2, 1, 3, 1, 2, 2, 5, 1, 2, 2, 4, 2, 4, 4, 9, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 17, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 33, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8

Offset: 0

Author

Paul Barry, Dec 11 2004

Keywords

• easy
• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A048896.

Programs

Mathematica

Table[Sum[Mod[Binomial[2n-2,k],2],{k,0,n}],{n,0,100}] (* Harvey P. Dale, Oct 20 2011 *)

Python

def A101691(n): return sum((not ~(n-1<<1)&k) for k in range(n+1)) # Chai Wah Wu, Jul 31 2025

Formula

a(n) = Sum_{k=0..n} mod(binomial(2n-2, k), 2).

a(2^n) = A094373(n). a(2^n-1) = 1,1,1,2,4,8,16,...