A135416 a(n) = A036987(n)*(n+1)/2.

1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

Keywords

• nonn

Comments

Guy Steele defines a family of 36 integer sequences, denoted here by GS(i,j) for 1 <= i, j <= 6, as follows. a[1]=1; a[2n] = i-th term of {0,1,a[n],a[n]+1,2a[n],2a[n]+1}; a[2n+1] = j-th term of {0,1,a[n],a[n]+1,2a[n],2a[n]+1}. The present sequence is GS(1,5).

The full list of 36 sequences:

GS(1,1) = A000007

GS(1,2) = A000035

GS(1,3) = A036987

GS(1,4) = A007814

GS(1,5) = A135416 (the present sequence)

GS(1,6) = A135481

GS(2,1) = A135528

GS(2,2) = A000012

GS(2,3) = A000012

GS(2,4) = A091090

GS(2,5) = A135517

GS(2,6) = A135521

GS(3,1) = A036987

GS(3,2) = A000012

GS(3,3) = A000012

GS(3,4) = A000120

GS(3,5) = A048896

GS(3,6) = A038573

GS(4,1) = A135523

GS(4,2) = A001511

GS(4,3) = A008687

GS(4,4) = A070939

GS(4,5) = A135529

GS(4,6) = A135533

GS(5,1) = A048298

GS(5,2) = A006519

GS(5,3) = A080100

GS(5,4) = A087808

GS(5,5) = A053644

GS(5,6) = A000027

GS(6,1) = A135534

GS(6,2) = A038712

GS(6,3) = A135540

GS(6,4) = A135542

GS(6,5) = A054429

GS(6,6) = A003817

(with a(0)=1): Moebius transform of A038712.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to binary expansion of n

Crossrefs

Equals A048298(n+1)/2. Cf. A036987, A182660.

Programs

Maple

GS:=proc(i,j,M) local a,n; a:=array(1..2*M+1); a[1]:=1;
for n from 1 to M do
a[2*n] :=[0,1,a[n],a[n]+1,2*a[n],2*a[n]+1][i];
a[2*n+1]:=[0,1,a[n],a[n]+1,2*a[n],2*a[n]+1][j];
od: a:=convert(a,list); RETURN(a); end;
GS(1,5,200):

Mathematica

i = 1; j = 5; Clear[a]; a[1] = 1; a[n_?EvenQ] := a[n] = {0, 1, a[n/2], a[n/2]+1, 2*a[n/2], 2*a[n/2]+1}[[i]]; a[n_?OddQ] := a[n] = {0, 1, a[(n-1)/2], a[(n-1)/2]+1, 2*a[(n-1)/2], 2*a[(n-1)/2]+1}[[j]]; Array[a, 105] (* Jean-François Alcover, Sep 12 2013 *)

PARI

A048298(n) = if(!n,0,if(!bitand(n,n-1),n,0));
A135416(n) = (A048298(n+1)/2); \\ Antti Karttunen, Jul 22 2018

Python

def A135416(n): return int(not(n&(n+1)))*(n+1)>>1 # Chai Wah Wu, Jul 06 2022

Formula

G.f.: sum{k>=1, 2^(k-1)*x^(2^k-1) }.

Recurrence: a(2n+1) = 2a(n), a(2n) = 0, starting a(1) = 1.

Extensions

Formulae and comments by Ralf Stephan, Jun 20 2014