A080100 a(n) = 2^(number of 0's in binary representation of n).

1, 1, 2, 1, 4, 2, 2, 1, 8, 4, 4, 2, 4, 2, 2, 1, 16, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1, 32, 16, 16, 8, 16, 8, 8, 4, 16, 8, 8, 4, 8, 4, 4, 2, 16, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1, 64, 32, 32, 16, 32, 16, 16, 8, 32, 16, 16, 8, 16, 8, 8, 4, 32, 16, 16, 8, 16, 8, 8, 4, 16, 8, 8, 4, 8, 4

Offset: 0

Author

Reinhard Zumkeller, Jan 28 2003

Keywords

• nonn
• base
• changed

Comments

Number of numbers k, 0<=k<=n, such that (k AND n) = 0 (bitwise logical AND): a(n) = #{k : T(n,k)=n, 0<=k<=n}, where T is defined as in A080099.

Same parity as the Catalan numbers (A000108). - Paul D. Hanna, Nov 14 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..8191
• George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
• Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Crossrefs

Cf. A001316.

Cf. A002487.

This is Guy Steele's sequence GS(5, 3) (see A135416).

Cf. A048896.

Cf. A023416, A080791.

Programs

Haskell

import Data.List (transpose)
a080100 n = a080100_list !! n
a080100_list =  1 : zs where
   zs =  1 : (concat $ transpose [map (* 2) zs, zs])
-- Reinhard Zumkeller, Aug 27 2014, Mar 07 2011

Maple

a:= n-> 2^add(1-i, i=Bits[Split](n)):
seq(a(n), n=0..93);  # Alois P. Heinz, Aug 18 2025

Mathematica

f[n_] := 2^DigitCount[n, 2, 0]; f[0] = 1; Array[f, 94, 0] (* Robert G. Wilson v *)

PARI

a(n)=if(n<1,n==0,(2-n%2)*a(n\2))

PARI

a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,x^2)*(2+x)-1); polcoeff(A,n))

Python

def A080100(n): return 1<Chai Wah Wu, Aug 18 2025

Formula

G.f. satisfies: F(x^2) = (1+F(x))/(x+2). - Ralf Stephan, Jun 28 2003

a(2n) = 2a(n), n>0. a(2n+1) = a(n). - Ralf Stephan, Apr 29 2003

a(n) = 2^A080791(n). a(n)=2^A023416(n), n>0.

a(n) = sum(k=0, n, C(n+k, k) mod 2). - Benoit Cloitre, Mar 06 2004

a(n) = sum(k=0, n, C(2n-k, n) mod 2). - Paul Barry, Dec 13 2004

G.f. satisfies: A(x) = Sum_{n>=0} [A(x)^n (mod 2)]*x^n, where A(x)^n (mod 2) reduces all coefficients modulo 2 to {0,1}. - Paul D. Hanna, Nov 14 2012

Extensions

Keyword base added by Rémy Sigrist, Jan 18 2018