A081667 a(n) = Fibonacci(binomial(n+2,2)).

1, 2, 8, 55, 610, 10946, 317811, 14930352, 1134903170, 139583862445, 27777890035288, 8944394323791464, 4660046610375530309, 3928413764606871165730, 5358359254990966640871840, 11825896447871834976429068427, 42230279526998466217810220532898

Offset: 0

Author

Paul Barry, Mar 26 2003

Keywords

• easy
• nonn

Comments

Diagonal of Fibonacci-Pascal triangle A045995.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..96
• Peter M. Chema, Illustration of first 12 terms on a square spiral
• T. Kotek, J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020 [math.CO], 2013.

Crossrefs

Cf. A000045, A000217, A033192, A045995, A054783.

Programs

Maple

with(combinat): seq(fibonacci((n^2-n)/2),n=2..16); # Zerinvary Lajos, May 18 2008
# second Maple program:
a:= n-> (<<0|1>, <1|1>>^((n+1)*(n+2)/2))[1, 2]:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 20 2017

Mathematica

Table[Fibonacci[Binomial[n+2,2]],{n,0,20}] (* Harvey P. Dale, Dec 03 2014 *)

Sage

[fibonacci(binomial(n,2)) for n in range(2, 17)] # Zerinvary Lajos, Nov 30 2009

Formula

a(n) = sqrt(5)2^(-n(n+3)/2)(sqrt(5)+1)^((n^2+3n+2)/2)/10 + sqrt(5)2^(-n(n + 3)/2)(sqrt(5)-1)^((n^2+3n+ 2)/2)(-1)^(n(n+3)/2)/10.

a(n) = A045995(n+2,2).

a(n) = A000045(A000217(n+1)). - Peter M. Chema, Sep 18 2016. See the name.

Extensions

Name edited by Michel Marcus, Sep 25 2016