A081005 a(n) = Fibonacci(4n+3) + 1, or Fibonacci(2n+1)*Lucas(2n+2).

3, 14, 90, 611, 4182, 28658, 196419, 1346270, 9227466, 63245987, 433494438, 2971215074, 20365011075, 139583862446, 956722026042, 6557470319843, 44945570212854, 308061521170130, 2111485077978051, 14472334024676222, 99194853094755498, 679891637638612259

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..500
• Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Crossrefs

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A344212.

Programs

GAP

List([0..30], n-> Fibonacci(4*n+3)+1); # G. C. Greubel, Jul 15 2019

Magma

[Fibonacci(4*n+3)+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+3)+1) od: # James Sellers, Mar 03 2003

Mathematica

Fibonacci[4Range[0,30]+3]+1 (* or *) LinearRecurrence[{8,-8,1}, {3,14,90}, 30] (* Harvey P. Dale, Jan 02 2013 *)

PARI

vector(30, n, n--; fibonacci(4*n+3)+1) \\ G. C. Greubel, Jul 15 2019

Sage

[fibonacci(4*n+3)+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.: (3-10*x+2*x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012

Product_{n>=0} (1 + 1/a(n)) = 1 + 1/sqrt(5) = A344212. - Amiram Eldar, Nov 28 2024

Extensions

More terms from James Sellers, Mar 03 2003