A007728 5th binary partition function.

1, 1, 2, 2, 4, 3, 5, 4, 8, 6, 9, 7, 12, 8, 12, 9, 17, 12, 18, 14, 23, 15, 22, 16, 28, 19, 27, 20, 32, 20, 29, 21, 38, 26, 38, 29, 47, 30, 44, 32, 55, 37, 52, 38, 60, 37, 53, 38, 66, 44, 63, 47, 74, 46, 66, 47, 79, 52, 72, 52, 81, 49, 70, 50, 88, 59, 85, 64

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• look

Comments

The number of ways of writing n as a sum of powers of 2, each power being used at most four times. - Dmitry Kamenetsky, Jul 14 2023

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.

Crossrefs

A column of A072170.

Cf. A000123, A018819.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<0, 0, add(`if`(n-j*2^i<0, 0,
         b(n-j*2^i, i-1)), j=0..4)))
    end:
a:= n-> b(n, ilog2(n)):
seq(a(n), n=0..70);  # Alois P. Heinz, Jun 21 2012

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 0, 0, Sum[If[n-j*2^i < 0, 0, b[n-j*2^i, i-1, k]], {j, 0, k-1}]]]; a[n_] := b[n, Log[2, n] // Floor, 5]; Table[a[n], {n, 0, 70} ] (* Jean-François Alcover, Jan 17 2014, after Alois P. Heinz *)

Formula

G.f.: Product_{k>=0} (1 - x^(5*2^k))/(1 - x^(2^k)). - Ilya Gutkovskiy, Jul 09 2019

Extensions

More terms from Vladeta Jovovic, May 06 2004