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,...).
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 01 2004
EXTENSIONS
More terms from Benoit Cloitre, Jul 02 2004
STATUS
approved