A096655 a(n) = F(n+1)*a(n-1) + F(n)*a(n-2), where F = A000045 (Fibonacci numbers), a(0)=1, a(1)=1.

1, 1, 3, 11, 64, 567, 7883, 172914, 6044619, 338333121, 30444101814, 4414062308985, 1032860468654721, 390416873200823322, 238543681049185056237, 235680767488198152732339

Offset

0,3

Comments

If the initial values are changed to a(0)=1 and a(1)=2, the resulting sequence (p(0),p(1),...)=(1,2,5,19,....) is essentially A089126. Writing A096655 as (q(0),q(1),...), the quotients p(n)/q(n) are the self-convergents to the number 1.719525... whose self-continued fraction is (1,1,2,3,5,...)=A000045. For definitions, see A096654. Now writing A096655 as (p(0),p(1),...) and A096656 as (q(0),q(1),...), the quotients p(n)/q(n) are the self-convergents to the number 1.389805... whose self-continued fraction is (1,2,3,5,...).

Links

Table of n, a(n) for n=0..15.

Formula

a(n) is asymptotic to c*phi^(n(n+1)/2)/5^(n/2) where c=3.487197183858494166192... and phi is the golden ratio. - Benoit Cloitre, Jul 02 2004

Example

a(2) = F(3)*1 + F(2)*1 = 3, a(3) = F(4)*3 + F(3)*1 = 11.

Mathematica

a[0] = 1; a[1] = 1; a[n_] := Fibonacci[n + 1]*a[n - 1] + Fibonacci[n]*a[n - 2]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Jul 09 2004 *)

Crossrefs

Cf. A000045, A089126, A096654.

Sequence in context: A326092 A292792 A199135 * A030226 A309174 A233099

Adjacent sequences: A096652 A096653 A096654 * A096656 A096657 A096658

Keyword

nonn

Author

Clark Kimberling, Jul 01 2004

Extensions

More terms from Benoit Cloitre, Jul 02 2004

Status

approved