OFFSET
0,3
COMMENTS
This sequence was suggested by Moshe Shmuel Newman. The idea came to him while reading a paper of Lev Shneerson.
Number of partitions of 2n into exactly n positive Fibonacci numbers. a(8) = 10: 82111111, 55111111, 53311111, 53221111, 52222111, 33331111, 33322111, 33222211, 32222221, 22222222. - Alois P. Heinz, Sep 18 2018
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..3500
FORMULA
From Alois P. Heinz, Sep 18 2018: (Start)
a(n) = [x^(2n) y^n] 1/Product_{j>=2} (1-y*x^A000045(j)).
a(n) = A319394(2n,n). (End)
EXAMPLE
a(5) = 4 because 4 of the 7 partitions of 5 have the required property: {5} {4,1} {3,2} {3,1,1}. The other 3 partitions of 5: {2,2,1} {2,1,1,1} and {1,1,1,1,1} each have an element which is < the sum of next two.
MAPLE
b:= proc(n, i, j) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n-i, min(n-i, i,
`if`(j=0, i, j-i)), i) +b(n, i-1, j)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Jul 29 2017
MATHEMATICA
b[n_, i_, j_]:=b[n, i, j]=If[n==0, 1, If[i<1, 0, b[n - i, Min[n - i, i, If[j==0, i, j - i]], i] + b[n, i - 1, j]]]; Table[b[n, n, 0], {n, 0, 60}] (* Indranil Ghosh, Aug 01 2017, after Maple code *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David S. Newman, May 11 2008
EXTENSIONS
Conjectured g.f. removed and a(0), a(35)-a(56) added by Alois P. Heinz, Jul 29 2017
STATUS
approved