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a(n+2) = F(n)*a(n+1) + F(n+1)*a(n) where F(n) = Fibonacci number (A000045) and a(0) = a(1) = 1.
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%I #15 Sep 29 2016 15:43:31

%S 1,1,1,2,4,14,62,422,4182,63228,1469976,53456724,3070947684,

%T 281012112132,41181274957380,9701178631343304,3682464921740427408,

%U 2255878665570796559928,2232433139398395667219512

%N a(n+2) = F(n)*a(n+1) + F(n+1)*a(n) where F(n) = Fibonacci number (A000045) and a(0) = a(1) = 1.

%F Recurrence; a(n+2) = F(n) a(n+1) + F(n+1) a(n) where F(n) = Fibonacci number (A000045)

%F a(n) = determinant of the tridiagonal matrix of order n+1, where the subdiagonal is [-1,-1,...-1], the diagonal is [1,1,F(0),F(1),...,F(n-2)], and the superdiagonal is [F(0),F(1),...,F(n-1)] [_Emanuele Munarini_, Apr 17 2013]

%t a[n_] := a[n] = Fibonacci[n-2]a[n-1] + Fibonacci[n-1]a[n-2] a[0] = 1 a[1] = 1

%t RecurrenceTable[{a[0]==a[1]==1,a[n+2]==Fibonacci[n] a[n+1]+ Fibonacci[ n+1] a[n]},a,{n,20}] (* _Harvey P. Dale_, Sep 29 2016 *)

%o (Maxima) F(i,j) := if i=1 and j=1 then 1 elseif i=j+1 then -1 elseif i=j then fib(i-3) elseif i=j-1 then fib(i-1) else 0;

%o M(n):=genmatrix(lambda([i,j], F(i,j)), n+1, n+1);

%o makelist(determinant(M(n)),n,0,12); [_Emanuele Munarini_, Apr 17 2013]

%K nonn

%O 0,4

%A _Emanuele Munarini_, Dec 05 2003