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

%I A227426 #20 May 24 2018 02:41:34
%S A227426 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,
%T A227426 136,156,180,207,238,272,311,355,404,460,522,592,670,758,855,965,1087,
%U A227426 1223,1373,1543,1728,1936,2166,2421,2702,3016,3359,3741,4162,4626,5136,5702,6320,7002,7753,8576,9479,10473
%N A227426 Number of partitions into distinct parts without three consecutive parts.
%C A227426 Number of partitions into distinct parts with maximal perimeter.
%C A227426 For n>=1, diagonal of A227344.
%H A227426 Joerg Arndt and Alois P. Heinz, <a href="/A227426/b227426.txt">Table of n, a(n) for n = 0..10000</a>
%F A227426 a(n) = c * exp(r*sqrt(n)) / n^(3/4), where r = 1.75931899568... and c = 0.2080626386... - _Vaclav Kotesovec_, May 24 2018
%p A227426 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A227426        b(n, i-1, 0)+`if`(i>n or t=2, 0, b(n-i, i-1, t+1))))
%p A227426     end:
%p A227426 a:= n-> b(n, n, 0):
%p A227426 seq(a(n), n=0..80);  # _Alois P. Heinz_, Jul 15 2013
%t A227426 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_ *)
%o A227426 (Haskell)
%o A227426 a227426 = p 1 1 where
%o A227426   p _ _ 0 = 1
%o A227426   p k i m = if m < k then 0 else p (k + i) (3 - i) (m - k) + p (k + 1) 1 m
%o A227426 -- _Reinhard Zumkeller_, Jul 14 2013
%Y A227426 Cf. A000009.
%K A227426 nonn
%O A227426 0,4
%A A227426 _Joerg Arndt_, Jul 11 2013