A094586 Central numbers of the triangle T of all positive differences of distinct Fibonacci numbers.

1, 5, 16, 47, 131, 356, 953, 2529, 6676, 17567, 46135, 121016, 317201, 831053, 2176712, 5700303, 14926171, 39081404, 102323209, 267896585, 701380076, 1836265535, 4807451951, 12586147632, 32951083681, 86267253461, 225850919488

Offset: 1

Author

Clark Kimberling, May 13 2004

Keywords

• nonn
• easy

Comments

As T is also the triangle of sums of consecutive distinct Fibonacci numbers, a(n) is such a sum, namely Sum_{j=n+1..2n} Fibonacci(j).

Examples

			a(4) = F(10)-F(6) = 55-8 = 47.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..2000
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Crossrefs

Cf. A000045, A094585.

Programs

GAP

List([1..30],n->Fibonacci(2*n+2)-Fibonacci(n+2)); # Muniru A Asiru, Apr 28 2019

Magma

F:=Fibonacci; [F(2*n+2)-F(n+2): n in [1..30]]; // G. C. Greubel, Jul 14 2019

Mathematica

Table[Sum[Fibonacci[n+i], {i,n}], {n,30}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{F=Fibonacci}, Table[F[2n+2]-F[n+2], {n,30}]] (* G. C. Greubel, Jul 14 2019 *)
LinearRecurrence[{4,-3,-2,1},{1,5,16,47},30] (* Harvey P. Dale, Dec 31 2024 *)

PARI

vector(30, n, f=fibonacci; f(2*n+2)-f(n+2)) \\ G. C. Greubel, Jul 14 2019

Sage

f=fibonacci; [f(2*n+2)-f(n+2) for n in (1..30)] # G. C. Greubel, Jul 14 2019

Formula

a(n) = Fibonacci(2n+2) - Fibonacci(n+2) = A094585(2n-1, n).

G.f.: x*(1+x-x^2)/((1-x-x^2)*(1-3*x+x^2)). - Colin Barker, Sep 16 2012