A185350 Number of parts in all partitions of n in which no part occurs more than twice.

0, 1, 3, 3, 8, 11, 17, 23, 36, 48, 69, 88, 125, 157, 212, 271, 356, 445, 574, 711, 906, 1118, 1400, 1711, 2125, 2583, 3171, 3828, 4666, 5604, 6777, 8095, 9730, 11567, 13815, 16357, 19429, 22910, 27077, 31801, 37432, 43802, 51338, 59871, 69914, 81273, 94562

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..11410 (terms 0..1000 from Alois P. Heinz)

Formula

a(n) = Sum_{k>=0} k*A209318(n,k).

a(n) ~ log(3) * exp(2*Pi*sqrt(n)/3) / (2*Pi*n^(1/4)). - Vaclav Kotesovec, May 26 2018

Example

a(6) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
a(7) = 23: [7], [6,1], [5,2], [4,3], [5,1,1], [4,2,1], [3,3,1], [3,2,2], [3,2,1,1].

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 2))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[b[n - i j, i - 1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]] j}, {j, 0, Min[n/i, k]} ] ] ];
a[n_] := b[n, n, 2][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)
Table[Length[Flatten[Select[IntegerPartitions[n], Max[Length/@Split[#]]<3&]]], {n, 0, 50}] (* Harvey P. Dale, Jul 04 2023 *)

Crossrefs

Column k=2 of A210485.

Cf. A015723, A117148, A209318.

Sequence in context: A368726 A052407 A233174 * A279910 A105039 A358834

Adjacent sequences: A185347 A185348 A185349 * A185351 A185352 A185353

Keyword

nonn

Author

Alois P. Heinz, Jan 21 2013

Status

approved