A143212 a(n) = Fibonacci(n) * (Fibonacci(n+2) - 1).

1, 2, 8, 21, 60, 160, 429, 1134, 2992, 7865, 20648, 54144, 141897, 371722, 973560, 2549421, 6675460, 17478176, 45761045, 119808150, 313668576, 821205937, 2149962768, 5628704256, 14736185425, 38579909330, 101003635304

Offset: 1

Author

Gary W. Adamson, Jul 30 2008

Keywords

• nonn

Comments

Lim_{n -> oo} a(n)/a(n-1) tends to phi^2.

a(n) = Product of sum of first n Fibonacci numbers and Fibonacci number(n). - Vladimir Joseph Stephan Orlovsky, Oct 13 2009

Examples

			a(5) = 60 = F(5) * (F(7)-1) = 5*12.
a(5) = 60 = sum of row 5 terms of triangle A143211: (5 + 5 + 10 + 15 + 25).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Crossrefs

Cf. A000045, A000071, A143211.

Programs

Magma

[Fibonacci(n)*(Fibonacci(n+2)-1): n in [1..40]]; // G. C. Greubel, Jul 21 2024

Mathematica

LinearRecurrence[{3,1,-5,-1,1}, {1,2,8,21,60}, 40] (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009 *)
Table[Fibonacci[n](Fibonacci[n+2]-1),{n,30}] (* Harvey P. Dale, Dec 14 2012 *)

SageMath

[fibonacci(n)*(fibonacci(n+2)-1) for n in range(1,41)] # G. C. Greubel, Jul 21 2024

Formula

a(n) = A000045(n) * A000071(n+2).

a(n) = Sum_{k=1..n} A143211(n, k) (row sums of A143211).

From R. J. Mathar, Sep 06 2008: (Start)

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

a(n) = (-5*A000045(n+1) + 3*(-1)^n + 7*A001906(n+1) -3*A001906(n))/5. (End)

a(n) = Fibonacci(n)*Sum_{k=0..n} Fibonacci(k). - Paul Barry, Jan 05 2009