A343944 Total number of parts in all partitions of n into powers of 2: p1 <= p2 <= ... <= p_k such that p_i <= 1 + Sum_{j=1..i-1} p_j.

0, 1, 2, 5, 7, 12, 15, 29, 35, 50, 58, 86, 98, 128, 143, 225, 251, 318, 350, 453, 495, 603, 653, 846, 914, 1092, 1172, 1419, 1517, 1773, 1886, 2521, 2687, 3130, 3322, 3917, 4147, 4759, 5021, 5909, 6227, 7082, 7442, 8537, 8955, 10076, 10544, 12326, 12898, 14452

Offset: 0

Author

Alois P. Heinz, May 04 2021

Keywords

• nonn

Examples

			a(5) = 12 = 5+4+3: [1,1,1,1,1], [1,1,1,2], [1,2,2].
a(6) = 15 = 6+5+4: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2].
a(7) = 29 = 7+6+5+4+4+3: [1,1,1,1,1,1,1], [1,1,1,1,1,2], [1,1,1,2,2], [1,2,2,2], [1,1,1,4], [1,2,4].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A281688, A343820.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<0, 0, (p-> `if`(
       p>n or p>n-p+1, 0, (h-> h+[0, h[1]])(b(n-p, i))))(2^i)+b(n, i-1)))
    end:
a:= n-> b(n, ilog2(n))[2]:
seq(a(n), n=0..60);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 0, {0, 0}, Function[p, If[p > n || p > n - p + 1, {0, 0}, Function[h, h + {0, h[[1]]}][b[n - p, i]]]][2^i] + b[n, i - 1]]];
a[n_] := b[n, Floor@Log2[n]][[2]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)