OFFSET
0,3
COMMENTS
If only the order of parts 1 and 2 matters, then the remaining parts can be frozen "[]" in a partition subsequence; e.g., a(15) would count the sequence 5,4,3,2,1 twice: [5,4,3]2,1 and [5,4,3]1,2. (Also see example.)
FORMULA
EXAMPLE
For n=6, the 21 sequences counted are [6]; [5],1; [4],2; [3,3], [4],1,1; [3],2,1; [3],1,2; 2,2,2; [3],1,1,1; 2,2,1,1; 2,1,2,1; 1,2,2,1; 1,2,1,2; 1,1,2,2; 2,1,1,2; 2,1,1,1,1; 1,2,1,1,1; 1,1,2,1,1; 1,1,1,2,1; 1,1,1,1,2; and 1,1,1,1,1,1.
MATHEMATICA
Table[PartitionsP[n] + Sum[Fibonacci[k] PartitionsP[n - 2 - k], {k, n - 2}], {n, 0, 50}] (* Indranil Ghosh, Jun 29 2017 *)
PROG
(PARI) a275388(n)=sum(k=1, n, fibonacci(k)*numbpart(n - k));
a(n)=numbpart(n)+a275388(n - 2); \\ Indranil Ghosh, Jun 29 2017
(Python)
from sympy import fibonacci, npartitions
def a(n): return npartitions(n) + sum([fibonacci(k)*npartitions(n - 2 - k) for k in range(1, n - 1)])
print([a(n) for n in range(51)]) # Indranil Ghosh, Jun 29 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Gregory L. Simay, Jun 29 2017
EXTENSIONS
More terms from Indranil Ghosh, Jun 29 2017
STATUS
approved