A131412 a(n) = n*(Fibonacci(n) - 1) + Fibonacci(n + 2) - 1.

1, 2, 7, 15, 32, 62, 117, 214, 385, 683, 1200, 2092, 3625, 6250, 10731, 18359, 31312, 53258, 90365, 152990, 258501, 435987, 734112, 1234200, 2072017, 3474002, 5817487, 9730719, 16258880, 27139478, 45258885, 75408742, 125538505, 208828475, 347119056, 576580804

Offset: 1

Author

Gary W. Adamson, Jul 08 2007

Keywords

• nonn

Examples

			a(4) = 15 = sum of row 4, triangle A131411; (3 + 3 + 4 + 5).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,0,-1).

Crossrefs

Row sums of A131411.

Cf. A000045, A131253.

Programs

GAP

F:=Fibonacci;; List([1..40], n-> F(n+2)+n*F(n)-(n+1)); # G. C. Greubel, Jul 13 2019

Magma

F:=Fibonacci; [F(n+2)+n*F(n)-(n+1): n in [1..40]]; // G. C. Greubel, Jul 13 2019

Mathematica

With[{F=Fibonacci}, Table[F[n+2]+n*F[n]-(n+1), {n,40}]] (* G. C. Greubel, Jul 13 2019 *)

PARI

a(n) = n*(fibonacci(n) - 1) + fibonacci(n+2) - 1; \\ Andrew Howroyd, Aug 10 2018

PARI

Vec((1 - 2*x + 3*x^2 - 3*x^3)/((1 - x)^2*(1 - x - x^2)^2) + O(x^40)) \\ Andrew Howroyd, Aug 10 2018

Sage

f=fibonacci; [f(n+2)+n*f(n)-(n+1) for n in (1..40)] # G. C. Greubel, Jul 13 2019

Formula

From Andrew Howroyd, Aug 10 2018: (Start)

a(n) = n*(Fibonacci(n) - 1) + Sum_{k=1..n} Fibonacci(k).

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6).

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

(End)

Extensions

Name changed and terms a(11) and beyond from Andrew Howroyd, Aug 10 2018