A081009 a(n) = Fibonacci(4n+3) - 1, or Fibonacci(2n+2)*Lucas(2n+1).

1, 12, 88, 609, 4180, 28656, 196417, 1346268, 9227464, 63245985, 433494436, 2971215072, 20365011073, 139583862444, 956722026040, 6557470319841, 44945570212852, 308061521170128, 2111485077978049, 14472334024676220, 99194853094755496

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), A134944.

Programs

GAP

List([0..30], n-> Fibonacci(4*n+3)-1); # G. C. Greubel, Jul 14 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[4*Range[0,30] +3] -1 (* G. C. Greubel, Jul 14 2019 *)
LinearRecurrence[{8,-8,1},{1,12,88},30] (* Harvey P. Dale, Sep 23 2019 *)

PARI

vector(30, n, n--; fibonacci(4*n+3)-1) \\ G. C. Greubel, Jul 14 2019

Sage

[fibonacci(4*n+3)-1 for n in (0..30)] # G. C. Greubel, Jul 14 2019

Formula

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

G.f.: (1+4*x)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012

Product_{n>=1} (1 - 1/a(n)) = (5+sqrt(5))/8 = A134944 + 1/2. - Amiram Eldar, Nov 28 2024

Extensions

More terms from James Sellers, Mar 03 2003