cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A206775 Values>0 taken by (-1)*A206774.

Original entry on oeis.org

1, 4, 6, 7, 11, 12, 15, 17, 22, 24, 28, 31, 40, 42, 43, 47
Offset: 1

Views

Author

Benoit Cloitre, Jan 10 2013

Keywords

Crossrefs

Cf. A206774.

A033922 Base-2 digital convolution sequence.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 3, 4, 4, 5, 4, 5
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

Definition: a(0) = 1; for n > 0, let the base-2 representation of n be 2^k_1 + ... + 2^k_i, then a(n) = a(k_1) + ... + a(k_i).
The sequence can be constructed as follows. Let r(n)=[x(1),x(2),...,x(2^n)] denote a run of 2^n elements. Then r(n+1) is a run of length 2^(n+1) defined as the concatenation of r(n) and [x(1)+x(n), x(2)+x(n), ..., x(2^n)+x(n)]. Letting x(1)=0 and x(2)=1 we get r(1)=[0,1], r(2)=[0, 1, 1, 2], r(3)=[0, 1, 1, 2, 1, 2, 2, 3], r(4)=[0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5], etc. Replacing the leading zero by 1 in r(infinity) we get A033922. - Benoit Cloitre, Jan 10 2013
Number of 0's in Goodstein base-2 hereditary representation of n. For example, a(266)=5 once that 266 = 2^(2^(2^(2^0)+2^0)) + 2^(2^(2^0)+2^0) + 2^(2^0). - Flávio V. Fernandes, Jul 22 2025

Examples

			For example, 6 = 2^2 + 2^1, so a(6) = a(2) + a(1) = 2.

Links

Crossrefs

Cf. A033639, A014221 (n such that a(n)=1), A206774 (first differences).
Cf. A048793, A133457; A348296; A000120.

Programs

  • Maple
    a:= proc(n) option remember; local c, m, t; if n=0 then 1 else m:= n; c:=0; for t from 0 while m<>0 do c:= c+ `if`(irem(m, 2, 'm')=1, a(t), 0) od; c fi end: seq(a(n), n=0..120);  # Alois P. Heinz, Jul 13 2011
  • PARI
    al(n)=local(v,k,e);v=vector(n+1);v[1]=1;for(m=1,n,k=m;e=0;while(k>0,if(k%2,v[m+1]+=v[e+1]);e++;k\=2));v /* Benoit Cloitre, Jan 10 2013 */
  • PARI
    /* to compute quickly 2^m terms of the sequence */ m=10;v=[0,1];for(n=2,m,v=concat(v,vector(2^n/2,i,v[i]+v[n])));a(n)=if(n<2,1,v[n]) /* Benoit Cloitre, Jan 16 2013 */

Formula

From Flávio V. Fernandes, Jul 31 2025: (Start)
a(n) = a(2^n).
a(n) = Sum_{k=1..A000120(n)} a(A048793(n,k)-1) for n >= 1. (End)

Extensions

Edited by Franklin T. Adams-Watters, Jul 13 2011
Showing 1-2 of 2 results.

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