A227614 Number of partitions of n into distinct parts with perimeter n-2.

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 8, 9, 11, 13, 14, 17, 19, 21, 25, 28, 32, 37, 42, 47, 55, 61, 69, 78, 88, 98, 112, 124, 140, 157, 176, 196, 221, 245, 274, 305, 340, 377, 420, 465, 517, 573, 634, 702, 777, 858, 949, 1047, 1154, 1273

Offset: 6

Author

Alois P. Heinz, Jul 17 2013

Keywords

• nonn

Comments

The perimeter is the sum of all parts having less than two neighbors.

a(n) counts all partitions of n into distinct parts where only part 2 has two neighbors.

Examples

			a(6) = 1: [1,2,3].
a(11) = 1: [1,2,3,5].
a(17) = 2: [1,2,3,5,6], [1,2,3,11].
a(19) = 3: [1,2,3,5,8], [1,2,3,6,7], [1,2,3,13].
a(21) = 4: [1,2,3,7,8], [1,2,3,5,10], [1,2,3,6,9], [1,2,3,15].
a(23) = 5: [1,2,3,5,12], [1,2,3,6,11], [1,2,3,7,10], [1,2,3,8,9], [1,2,3,17].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 6..1000

Crossrefs

Cf. A227344.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<5, 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-6, n-6, 0):
seq(a(n), n=6..100);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n==0, 1, If[i<5, 0, b[n, i-1, 0] + If[i>n || t==2, 0, b[n-i, i-1, t+1]]]]; a[n_] := b[n-6, n-6, 0]; Table[a[n], {n, 6, 100}] (* Jean-François Alcover, Feb 17 2017, translated from Maple *)

Formula

a(n) = A227344(n,n-2).