OFFSET
0,3
FORMULA
Identity: a(n)*a(n+1)*a(n+4) + a(n)*a(n+2)^2 + a(n+1)^2*a(n+2) -
a(n)*a(n+1)*a(n+3) - a(n)*a(n+2)*a(n+3) - a(n+1)*a(n+2)^2 = 0.
a(n) = det(M(n)), where M(n) is the n x n tridiagonal matrix whose entries m(i,j) are defined as follows: m(i,i) = 1, m(i,i-1) = -1, m(i,i+1) = Lucas(i) = A000032(i) and m(i,j) = 0 otherwise (for i, j = 1..n).
a(n) ~ c * ((1 + sqrt(5))/2)^(n^2/4), where c = 3.937032778079679557806160201647101521427287177807702744719421167... if n is even and c = 4.036450637503687376356038529840104507940244677583731628506054362... if n is odd. - Vaclav Kotesovec, Feb 19 2016
MATHEMATICA
Lucas[n_] := Fibonacci[n-1] + Fibonacci[n+1]
a[n_] := a[n] = a[n-1] + Lucas[n-1] a[n - 2]
a[0] = 1;
a[1] = 1;
Table[a[n], {n, 0, 100}]
PROG
(Maxima) lucas(n) := fib(n-1) + fib(n+1);
a[0]: 1$
a[1]: 1$
a[n] := a[n-1] + lucas(n-1)*a[n-2]$
makelist(a[n], n, 0, 40);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Feb 19 2016
STATUS
approved