A269068 - OEIS

A269068

a(n+2) = a(n+1) + L(n+1)*a(n), where L = Lucas number (A000032) and a(0) = a(1) = 1.

1, 1, 2, 5, 13, 48, 191, 1055, 6594, 56179, 557323, 7467340, 118374617, 2522858097, 64196033554, 2190965409325, 89754355176981, 4925215013557256, 325438017350556407, 28783330365684381575, 3071303354576036230618, 438476741796283676315643, 75611697648399346456921811, 17440606103006621779585331540

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..23.

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);