A387736 Number of partitions of n such that the largest part is >= n/2, adjacent parts differ by at least 2, and the smallest part >= 2.

0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 8, 11, 11, 14, 14, 18, 18, 22, 22, 28, 28, 34, 34, 42, 42, 51, 51, 62, 62, 74, 74, 89, 89, 105, 105, 125, 125, 147, 147, 173, 173, 202, 202, 237, 237, 275, 275, 320, 320, 370, 370, 428, 428, 492, 492, 567, 567, 649

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from Sean A. Irvine)

Formula

a(n) = Sum_{k=ceiling(n/2)..n} b(n-k,k-2) = Sum_{k=0..floor(n/2)} b(k,n-k-2) where b(0,k)=1, b(n,k) = b(n,k-1) + b(n-k, k-2) and b(n,k)=0 if n<0 or k<2.

a(2*n+1) = a(2*n).

Example

a(6)=2 because of the partitions 6 and 4+2.

Mathematica

b[n_, k_]:=If[n==0, 1, If[n<0||k<2, 0, b[n, k-1]+b[n-k, k-2]]]; a[n_]:=Sum[b[n-k, k-2], {k, Ceiling[n/2], n}]; Join[{0}, Array[a, 61, 2]] (* James C. McMahon, Sep 08 2025 *)

Crossrefs

Cf. A387735, A387737, A387738, A387739, A387740.

Sequence in context: A034322 A362612 A319169 * A050365 A029026 A003106

Adjacent sequences: A387733 A387734 A387735 * A387737 A387738 A387739

Keyword

nonn

Author

Sean A. Irvine, Sep 06 2025

Extensions

a(0)=0 prepended by Alois P. Heinz, Sep 08 2025

Status

approved