login
A318756
Total number of binary digits used to write all partitions of n in binary notation.
3
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
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
Sequence in context: A307817 A210433 A182099 * A365675 A195334 A009918
KEYWORD
nonn,base
AUTHOR
David S. Newman, Sep 02 2018
STATUS
approved