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

1,2

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A066624, A070939.

Sequence in context: A307817 A210433 A182099 * A365675 A195334 A009918

Adjacent sequences: A318753 A318754 A318755 * A318757 A318758 A318759

Keyword

nonn,base

Author

David S. Newman, Sep 02 2018

Status

approved