A318756 Total number of binary digits used to write all partitions of n in binary notation.

1, 4, 8, 18, 30, 55, 85, 141, 211, 324, 467, 691, 968, 1377, 1898, 2631, 3554, 4830, 6425, 8578, 11272, 14819, 19243, 25005, 32133, 41279, 52585, 66907, 84512, 106636, 133685, 167377, 208439, 259145, 320696, 396251, 487532, 598881, 732990, 895627, 1090752

Offset: 1

Author

David S. Newman, Sep 02 2018

Keywords

• nonn
• base

Examples

			For n = 3 there are 3 partitions which when written in binary are: 11, 10+1, 1+1+1, for a total of 8 binary integers.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A066624, A070939.

Programs

Maple

h:= proc(n) option remember; 1+ilog2(n) end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, b(n, i-1)+(p-> p+[0, p[1]*
        h(i)])(b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 27 2018

Mathematica

h[n_] := h[n] = 1 + Log[2, n] // Floor;
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, 0, b[n, i - 1] + Function[p, p + {0, p[[1]]*h[i]}][b[n - i, Min[n - i, i]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[1, 60] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz *)
Table[Length[Flatten[IntegerDigits[#,2]&/@IntegerPartitions[n]]],{n,50}] (* Harvey P. Dale, Aug 14 2021 *)

PARI

a(n)={subst(deriv(polcoef(1/prod(k=1, n, 1 - x^k*y^(logint(k,2) + 1) + O(x*x^n)), n)), y, 1)} \\ Andrew Howroyd, Sep 07 2018