A033639 -id:A033639 - cp's OEIS frontend

A033922 Base-2 digital convolution sequence.

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

David W. Wilson

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

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

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Benoit Cloitre, Plot of a(n) for n=1 up to 100000

Cf. A033639, A014221 (n such that a(n)=1), A206774 (first differences).

Cf. A048793, A133457; A348296; A000120.

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

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 */

/* 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 */

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)

Edited by Franklin T. Adams-Watters, Jul 13 2011

A033640 Base 3 digital convolution sequence.

Original entry on oeis.org

1, 1, 2, 1, 3, 7, 6, 20, 52, 6, 26, 104, 32, 162, 460, 356, 1438, 4048, 712, 3588, 15272, 5012, 27460, 90476, 64944, 300816, 912472, 90476, 155420, 611656, 1067892, 1770024, 4763360, 4151704, 14746316, 39566064, 8915064, 27813084, 109938548, 76294212, 222960908

Offset: 0

David W. Wilson

			Suppose base = 3 and a(0)..a(13) are 1 1 2 1 3 7 6 20 52 6 26 104 32 162. In base 3, 14 = 112, so we convolve the last three terms with 1, 1, 2 to obtain 104*1+32*1+162*2 = 460.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A033639, A033641, A033642, A033643, A033644, A033645, A033646, A033647.

a:= proc(n) option remember; `if`(n=0, 1, (l->
      add(l[i]*a(n-i), i=1..nops(l)))(convert(n, base, 3)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 14 2021

A033647 Base 10 digital convolution sequence.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 40320, 403200, 846720, 2943360, 12620160, 66044160, 408885120, 2928240000, 23834805120, 217441486080, 47669610240, 482552582400, 1060444385280, 4146438320640, 18706642053120, 101826086906880, 648369805547520

Offset: 0

David W. Wilson

			Suppose base = 3 and a(0)..a(13) are 1 1 2 1 3 7 6 20 52 6 26 104 32 162. In base 3, 14 = 112, so we convolve the last three terms with 1, 1, 2 to obtain 104*1+32*1+162*2 = 460.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A033639, A033640, A033641, A033642, A033643, A033644, A033645, A033646.

a:= proc(n) option remember; `if`(n=0, 1, (l->
      add(l[i]*a(n-i), i=1..nops(l)))(convert(n, base, 10)))
    end:
seq(a(n), n=0..33);  # Alois P. Heinz, Apr 14 2021

A260956 a(0)=1; a(n) = Sum_{k=1..n-1} d(k)*a(n-k), where d(m) is m-th bit in binary expansion of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 10, 10, 13, 23, 43, 66, 122, 231, 462, 462, 528, 759, 1452, 1980, 3201, 5412, 9603, 15015, 26598, 47025, 88638, 162261, 312939, 610863, 1221726, 1221726, 1310364, 1623303, 2547105, 3768831, 6300921, 9234588, 14715360, 21016281, 32797974

Offset: 0

Anders Hellström, Sep 10 2015

Alois P. Heinz, Table of n, a(n) for n = 0..4575 (terms n = 1..1000 from Anders Hellström)

Cf. A007088, A000120, A323355.

Cf. A033639 (when binary digits are taken in the reverse order).

a:= proc(n) option remember; `if`(n=0, 1, (l-> add(
      a(n-i)*l[-i], i=1..nops(l)))(convert(n, base, 2)))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 18 2019

a[0] = 1; a[n_] := a[n] = Total[(d = IntegerDigits[n, 2]) * Table[a[n - k], {k, 1, Length[d]}]]; Array[a, 50, 0] (* Amiram Eldar, Jul 25 2023 *)

first(m)=my(v=vector(m));v[1]=1;v[2]=1;for(i=3,m,v[i]=0;d=digits(i,2);for(j=1,#d,v[i]+=d[j]*v[i-j]));v

lista(nn) = {my(va = vector(nn), vb); va[1] = 1; for (n=2, nn, vb = binary(n); va[n] = sum(k=1, #vb, vb[k]*(if (n==k, 1, va[n-k])));); concat(1, va);} \\ Michel Marcus, Jan 12 2019

a(0)=1 prepended by Alois P. Heinz, Jan 18 2019

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A033922 Base-2 digital convolution sequence.

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A033640 Base 3 digital convolution sequence.

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A033647 Base 10 digital convolution sequence.

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A260956 a(0)=1; a(n) = Sum_{k=1..n-1} d(k)*a(n-k), where d(m) is m-th bit in binary expansion of n.

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