A081003 a(n) = Fibonacci(4n+1) + 1, or Fibonacci(2n+1)*Lucas(2n).

2, 6, 35, 234, 1598, 10947, 75026, 514230, 3524579, 24157818, 165580142, 1134903171, 7778742050, 53316291174, 365435296163, 2504730781962, 17167680177566, 117669030460995, 806515533049394, 5527939700884758, 37889062373143907, 259695496911122586

Offset: 0

Author

R. K. Guy, Mar 01 2003

Keywords

• nonn
• easy

References

• Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Crossrefs

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A296182.

Programs

GAP

List([0..30], n-> Fibonacci(4*n+1)+1); # G. C. Greubel, Jul 15 2019

Magma

[Fibonacci(4*n+1) +1: n in [0..30]]; // Vincenzo Librandi, Apr 15 2011

Maple

with(combinat): for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+1)+1) od:  # James Sellers, Mar 03 2003

Mathematica

Fibonacci[4*Range[0,30]+1]+1 (* or *) LinearRecurrence[{8,-8,1}, {2,6,35}, 30] (* Harvey P. Dale, Jul 20 2011 *)

PARI

vector(30, n, n--; fibonacci(4*n+1)+1) \\ G. C. Greubel, Jul 15 2019

Sage

[fibonacci(4*n+1)+1 for n in (0..30)] # G. C. Greubel, Jul 15 2019

Formula

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).

G.f.: (2-10*x+3*x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012

Product_{n>=0} (1 + 1/a(n)) = (2 + phi)/2 (A296182). - Amiram Eldar, Nov 28 2024

Extensions

More terms from James Sellers, Mar 03 2003