A305412 a(n) = F(n)*F(n+1) + F(n+2), where F = A000045 (Fibonacci numbers).

1, 3, 5, 11, 23, 53, 125, 307, 769, 1959, 5039, 13049, 33929, 88451, 230957, 603667, 1578823, 4130829, 10810469, 28295411, 74067401, 193893263, 507590495, 1328842801, 3478880593, 9107706243, 23844088085, 62424315227, 163428464759, 427860443429, 1120151837069

Offset: 0

Author

Vincenzo Librandi, Jun 05 2018

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Crossrefs

Cf. A001654, A269803.

Cf. A059769: F(n)*F(n+1) - F(n+2), with offset 3.

Equals A000045 + A286983.

First differences are listed in A059727 (after 0).

Programs

GAP

List([0..35], n -> Fibonacci(n)*Fibonacci(n+1)+Fibonacci(n+2)); # Muniru A Asiru, Jun 06 2018

Magma

[Fibonacci(n)*Fibonacci(n+1)+Fibonacci(n+2): n in [0..30]];

Mathematica

Table[Fibonacci[n] Fibonacci[n+1] + Fibonacci[n+2], {n, 0, 30}]

Formula

G.f.: (1 - 5*x^2 - 2*x^3 + x^4)/((x + 1)*(1 - 3*x + x^2)*(1 - x - x^2)).

a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5).

5*a(n) = (-1)^(n+1) +5*F(n+2) + A002878(n). - R. J. Mathar, Nov 14 2019