A049686 a(n) = Fibonacci(8n)/3.

0, 7, 329, 15456, 726103, 34111385, 1602508992, 75283811239, 3536736619241, 166151337293088, 7805576116155895, 366695926122033977, 17226902951619441024, 809297742799991694151, 38019767008647990184073, 1786119751663655546957280, 83909608561183162716808087, 3941965482623944992143022809

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

a(n) = (t(i+4n) - t(i))/(t(i+2n+1) - t(i+2n-1)), where (t) is any sequence of the form t(n+2) = 8t(n+1) - 8t(n) + t(n-1) or of the form t(n+1) = 7t(n) - t(n-1) without regard to initial values as long as t(i+2n+1) - t(i+2n-1) != 0. - Klaus Purath, Jun 23 2024

Examples

			a(2) = F(8 * 2) / 3 = F(16) / 3 = 987 / 3 = 329. - _Indranil Ghosh_, Feb 05 2017
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..597
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (47,-1).

Crossrefs

Cf. A000045, A004187, A049668.

Programs

GAP

List([0..20], n-> Fibonacci(8*n)/3 ); # G. C. Greubel, Dec 14 2019

Magma

[Fibonacci(8*n)/3: n in [0..20]]; // G. C. Greubel, Dec 14 2019

Maple

with(combinat); seq( fibonacci(8*n)/3, n=0..20); # G. C. Greubel, Dec 14 2019

Mathematica

Fibonacci[8(Range[20]-1)]/3 (* G. C. Greubel, Dec 14 2019 *)
LinearRecurrence[{47,-1},{0,7},20] (* Harvey P. Dale, Dec 27 2019 *)

PARI

a(n) = fibonacci(8*n)/3; \\ Michel Marcus, Feb 05 2017

Sage

[fibonacci(8*n)/3 for n in (0..20)] # G. C. Greubel, Dec 14 2019

Formula

a(n) = 47*a(n-1) - a(n-2), n>1. a(0)=0, a(1)=7.

G.f.: 7*x/(1-47*x+x^2).

a(n) = A004187(2n).

a(n) = 7*A049668(n). - R. J. Mathar, Oct 26 2015

E.g.f.: 2*exp(47*x/2)*sinh(21*sqrt(5)*x/2)/(3*sqrt(5)). - Stefano Spezia, Dec 14 2019

Extensions

Better description and more terms from Michael Somos