A190173 a(n) = Sum_{1 <= i < j <= n} F(i)*F(j), where F(k) is the k-th Fibonacci number.

0, 1, 5, 17, 52, 148, 408, 1101, 2937, 7777, 20504, 53912, 141520, 371113, 972573, 2547825, 6672876, 17473996, 45754280, 119797205, 313650865, 821177281, 2149916400, 5628629232, 14736064032, 38579712913, 101003317493, 264430632401, 692289215332, 1812438042052

Offset

1,3

Links

Vincenzo Librandi and Bruno Berselli, Table of n, a(n) for n = 1..1000 (First 211 terms from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (4,-2,-6,4,2,-1).

Formula

a(n) = F(n+1)^2 - F(n+2) + (1-(-1)^n)/2.

G.f.: x^2*(1+x-x^2)/((1-x)*(1+x)*(1-x-x^2)*(1-3*x+x^2)). - Bruno Berselli, Jun 20 2012

Example

a(4) = F(1)*F(2) + F(1)*F(3) + F(1)*F(4) + F(2)*F(3) + F(2)*F(4) + F(3)*F(4) = 1 + 2 + 3 + 2 + 3 + 6 = 17.

Maple

with(combinat): seq(fibonacci(n+1)^2-fibonacci(n+2)+1/2-(1/2)*(-1)^n, n = 1 .. 30);

Mathematica

Table[Fibonacci[n + 1]^2 - Fibonacci[n + 1] + (1 - (-1)^n)/2, {n, 1, 50}] (* G. C. Greubel, Mar 04 2017 *)

Prog

(Magma) [Fibonacci(n+1)^2 - Fibonacci(n+2) + (1-(-1)^n)/2: n in [1..30]]; // Vincenzo Librandi, Jun 05 2011
(PARI) a(n)=fibonacci(n+1)^2-fibonacci(n+2)+n%2 \\ Charles R Greathouse IV, Jun 08 2011

Crossrefs

Cf. A000045.

Sequence in context: A146814 A034335 A337033 * A187257 A290186 A178703

Adjacent sequences: A190170 A190171 A190172 * A190174 A190175 A190176

Keyword

nonn,easy

Author

Emeric Deutsch, May 31 2011

Status

approved