A033639 Base-2 digital convolution sequence.

1, 1, 1, 2, 1, 2, 3, 6, 1, 3, 4, 13, 4, 11, 21, 49, 13, 17, 24, 62, 66, 103, 145, 338, 128, 297, 376, 1156, 763, 1564, 2592, 6451, 376, 1532, 1139, 4235, 4124, 11714, 8735, 26105, 5263, 21212, 18122, 77153, 35210, 100649, 135748, 369972, 95275, 207638

Offset: 0

Author

David W. Wilson

Keywords

• base
• nonn

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..5000
• Rémy Sigrist, PARI program for A033639.

Crossrefs

Cf. A033640, A033641, A033642, A033643, A033644, A033645, A033646, A033647, A033666, A033922, A260956.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, (l->
      add(l[i]*a(n-i), i=1..nops(l)))(Bits[Split](n)))
    end:
seq(a(n), n=0..49);  # Alois P. Heinz, Apr 14 2021

Mathematica

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

PARI

See Links section.