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

0,3

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.

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

Mathematica

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

Prog

(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))
(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

Crossrefs

Cf. A001316.

Cf. A002487.

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

Cf. A048896.

Cf. A023416, A080791.

Sequence in context: A109090 A220780 A323915 * A365425 A365381 A161822

Adjacent sequences: A080097 A080098 A080099 * A080101 A080102 A080103

Keyword

nonn,base

Author

Reinhard Zumkeller, Jan 28 2003

Extensions

Keyword base added by Rémy Sigrist, Jan 18 2018

Status

approved