OFFSET
0,4
COMMENTS
Equivalently, antidiagonal sums of the third quadrant of array A(k,m).
It seems that: a(n+1) is the sum of the n-th antidiagonal of triangle A101494; a(n)-(n mod 2) is the sum of the n-th antidiagonal of array A172236; and a(n+1)+(n mod 2) is the sum of row n of triangle A157103. - Mathew Englander, Feb 28 2021
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..600
Wikipedia, Fibonacci polynomials
Wikipedia, Quadrant (plane geometry)
FORMULA
a(n) = Sum_{j=0..n} F_j(n-j).
a(n+1) = Sum_{j = 0..n} Sum_{i = j..floor((n+j)/2)} binomial(i,j)*(n+j-2*i)^j (empirically). - Mathew Englander, Feb 28 2021
MAPLE
F:= (n, k)-> (<<0|1>, <1|k>>^n)[1, 2]:
a:= n-> add(F(j, n-j), j=0..n):
seq(a(n), n=0..30);
# second Maple program:
F:= proc(n, k) option remember;
`if`(n<2, n, k*F(n-1, k)+F(n-2, k))
end:
a:= n-> add(F(j, n-j), j=0..n):
seq(a(n), n=0..30);
# third Maple program:
a:= n-> add(combinat[fibonacci](j, n-j), j=0..n):
seq(a(n), n=0..30);
MATHEMATICA
a[n_] := Sum[Fibonacci[j, n - j], {j, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 02 2018, from 3rd Maple program *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 11 2018
STATUS
approved