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

1,2

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).

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).

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

Example

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

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 *)

Prog

(GAP) List([1..30], n->Fibonacci(2*n+2)-Fibonacci(n+2)); # Muniru A Asiru, Apr 28 2019
(PARI) vector(30, n, f=fibonacci; f(2*n+2)-f(n+2)) \\ G. C. Greubel, Jul 14 2019
(Magma) F:=Fibonacci; [F(2*n+2)-F(n+2): n in [1..30]]; // 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

Crossrefs

Cf. A000045, A094585.

Sequence in context: A086750 A086749 A194541 * A353133 A140336 A197201

Adjacent sequences: A094583 A094584 A094585 * A094587 A094588 A094589

Keyword

nonn,easy

Author

Clark Kimberling, May 13 2004

Status

approved