A332719 Index position of {n}^3 within the list of partitions of 3n in canonical ordering.

1, 3, 8, 19, 44, 93, 187, 357, 657, 1166, 2015, 3393, 5594, 9044, 14378, 22501, 34734, 52931, 79735, 118823, 175337, 256347, 371606, 534377, 762721, 1080979, 1521925, 2129330, 2961580, 4096006, 5634855, 7712558, 10505457, 14243772, 19227383, 25845241, 34600673

Offset

0,2

Comments

The canonical ordering of partitions is described in A080577.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Wikipedia, Integer Partition

Formula

a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*Pi*sqrt(n)). - Vaclav Kotesovec, Feb 28 2020

Example

a(2) = 8, because 222 has position 8 within the list of partitions of 6 in canonical ordering: 6, 51, 42, 411, 33, 321, 3111, 222, ... .

Maple

b:= proc(n, i) option remember;
     `if`(n=0, 1, b(n-i, i)+g(n, i-1))
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
     `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
    end:
a:= n-> g(3*n$2)-b(3*n, n)+1:
seq(a(n), n=0..37);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, b[n - i, i] + g[n, i - 1]];
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, If[i < 1, 0, g[n - i, Min[n - i, i]] + g[n, i - 1]]];
a[n_] := g[3n, 3n] - b[3n, n] + 1;
a /@ Range[0, 37] (* Jean-François Alcover, Jan 06 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A080577, A322761, A330661, A332706, A332720.

Sequence in context: A371796 A191787 A347310 * A326599 A121551 A189391

Adjacent sequences: A332716 A332717 A332718 * A332720 A332721 A332722

Keyword

nonn

Author

Alois P. Heinz, Feb 20 2020

Status

approved