A227426 Number of partitions into distinct parts without three consecutive parts.

1, 1, 1, 2, 2, 3, 3, 5, 6, 7, 9, 11, 13, 16, 20, 23, 28, 33, 39, 46, 55, 63, 75, 87, 101, 117, 136, 156, 180, 207, 238, 272, 311, 355, 404, 460, 522, 592, 670, 758, 855, 965, 1087, 1223, 1373, 1543, 1728, 1936, 2166, 2421, 2702, 3016, 3359, 3741, 4162, 4626, 5136, 5702, 6320, 7002, 7753, 8576, 9479, 10473

Offset: 0

Author

Joerg Arndt, Jul 11 2013

Keywords

• nonn

Comments

Number of partitions into distinct parts with maximal perimeter.

For n>=1, diagonal of A227344.

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000009.

Programs

Haskell

a227426 = p 1 1 where
  p   0 = 1
  p k i m = if m < k then 0 else p (k + i) (3 - i) (m - k) + p (k + 1) 1 m
-- Reinhard Zumkeller, Jul 14 2013

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1, 0)+`if`(i>n or t=2, 0, b(n-i, i-1, t+1))))
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Jul 15 2013

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n==0, 1, If[i<1, 0, b[n, i-1, 0] + If[i>n || t==2, 0, b[n-i, i-1, t+1]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 02 2015, after Alois P. Heinz *)

Formula

a(n) = c * exp(r*sqrt(n)) / n^(3/4), where r = 1.75931899568... and c = 0.2080626386... - Vaclav Kotesovec, May 24 2018