A099015 a(n) = Fib(n+1)*(2*Fib(n)^2 + Fib(n)*Fib(n-1) + Fib(n-1)^2).

1, 2, 8, 33, 140, 592, 2509, 10626, 45016, 190685, 807764, 3421728, 14494697, 61400482, 260096680, 1101787113, 4667245276, 19770767984, 83750317589, 354772037730, 1502838469496, 6366125914117, 26967342128548

Offset

0,2

Comments

Form the matrix A=[1,1,1,1;3,2,1,0;3,1,0,0;1,0,0,0]=(binomial(3-j,i)). Then a(n)=(2,2)-element of A^n.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..172 from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Formula

G.f.: (1-x-4*x^2)/((1+x-x^2)*(1-4*x-x^2)).

G.f.: (1-x-4*x^2)/(1-3*x-6*x^2+3*x^3+x^4).

a(n) = (3*Fib(3*n+1) + (-1)^n*Fib(n-3))/5.

a(n) = (2+sqrt(5))^n*(3/10 + 3*sqrt(5)/50) + (2-sqrt(5))^n*(3/10 - 3*sqrt(5)/50) + (-1)^n*((1/2 - sqrt(5)/2)^n*(1/5 + 2*sqrt(5)/25) + (1/5 - 2*sqrt(5)/25)*(1/2 + sqrt(5)/2)^n).

Mathematica

LinearRecurrence[{3, 6, -3, -1}, {1, 2, 8, 33}, 30] (* Harvey P. Dale, Nov 28 2015 *)
CoefficientList[Series[(1-x-4*x^2)/((1+x-x^2)*(1-4*x-x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 31 2017 *)

Prog

(Magma) [Fibonacci(n+1)*(2*Fibonacci(n)^2 + Fibonacci(n)*Fibonacci(n-1) + Fibonacci(n-1)^2): n in [0..30]]; // Vincenzo Librandi, Jun 05 2011
(PARI) a(n)=my(e=fibonacci(n-1), f=fibonacci(n)); (e+f)*(2*f^2+f*e+e^2) \\ Charles R Greathouse IV, Jun 05, 2011
(PARI) first(n) = Vec((1 - x - 4*x^2)/(1 - 3*x - 6*x^2 + 3*x^3 + x^4) + O(x^n)) \\ Iain Fox, Dec 31 2017

Crossrefs

Cf. A000045, A056570, A066258, A066259, A099014, A033887.

Sequence in context: A037716 A373752 A279014 * A150865 A150866 A150867

Adjacent sequences: A099012 A099013 A099014 * A099016 A099017 A099018

Keyword

easy,nonn

Author

Paul Barry, Sep 22 2004

Status

approved