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

0,4

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

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A000009.

Sequence in context: A097450 A062303 A180682 * A229950 A050318 A130841

Adjacent sequences: A227423 A227424 A227425 * A227427 A227428 A227429

Keyword

nonn

Author

Joerg Arndt, Jul 11 2013

Status

approved