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A269068 a(n+2) = a(n+1) + L(n+1)*a(n), where L = Lucas number (A000032) and a(0) = a(1) = 1. 1

%I

%S 1,1,2,5,13,48,191,1055,6594,56179,557323,7467340,118374617,

%T 2522858097,64196033554,2190965409325,89754355176981,4925215013557256,

%U 325438017350556407,28783330365684381575,3071303354576036230618,438476741796283676315643,75611697648399346456921811,17440606103006621779585331540

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

%F Identity: a(n)*a(n+1)*a(n+4) + a(n)*a(n+2)^2 + a(n+1)^2*a(n+2) -

%F a(n)*a(n+1)*a(n+3) - a(n)*a(n+2)*a(n+3) - a(n+1)*a(n+2)^2 = 0.

%F 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).

%F 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

%t Lucas[n_] := Fibonacci[n-1] + Fibonacci[n+1]

%t a[n_] := a[n] = a[n-1] + Lucas[n-1] a[n - 2]

%t a[0] = 1;

%t a[1] = 1;

%t Table[a[n], {n, 0, 100}]

%o (Maxima) lucas(n) := fib(n-1) + fib(n+1);

%o a[0]: 1$

%o a[1]: 1$

%o a[n] := a[n-1] + lucas(n-1)*a[n-2]$

%o makelist(a[n], n, 0, 40);

%Y Cf. A000032, A089125.

%K nonn

%O 0,3

%A _Emanuele Munarini_, Feb 19 2016

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Last modified May 7 02:20 EDT 2021. Contains 343636 sequences. (Running on oeis4.)