A081002 a(n) = Fibonacci(4n) + 1, or Fibonacci(2n-1)*Lucas(2n+1).

1, 4, 22, 145, 988, 6766, 46369, 317812, 2178310, 14930353, 102334156, 701408734, 4807526977, 32951280100, 225851433718, 1548008755921, 10610209857724, 72723460248142, 498454011879265, 3416454622906708, 23416728348467686

Offset

0,2

Comments

If x=a(n), y=a(n+1), z=a(n+2) or x=a(n+2), y=a(n+1), z=a(n), then x^2 -9*y*x +7*x*z +9*y^2 -9*z*y +z^2 = -45. - Alexander Samokrutov, Jul 02 2015

References

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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

Formula

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

From Paul Barry, Jan 20 2005: (Start)

G.f.: (1-4*x-2*x^2)/(1-8*x+8*x^2-x^3).

a(n) = Sum_{k=0..n} binomial(2n-k, 2k)*2^(2n-3k).

a(n) = 1 - 0^n + Sum_{k=0..2n} binomial(4n-k-1, k). (End)

Maple

with(combinat): for n from 0 to 30 do printf(`%d, `, fibonacci(4*n)+1) od: # James A. Sellers, Mar 01 2003

Mathematica

Fibonacci[4*Range[0, 30]]+1 (* or *) LinearRecurrence[{8, -8, 1}, {1, 4, 22}, 30] (* Harvey P. Dale, Feb 26 2015 *)

Prog

(Magma) [Fibonacci(4*n)+1: n in [0..30]]; // Vincenzo Librandi, Apr 20 2011
(PARI) Vec((1-4*x-2*x^2)/((1-x)*(1-7*x+x^2)) + O(x^30)) \\ Colin Barker, Dec 23 2014
(Sage) [fibonacci(4*n)+1 for n in (0..30)] # G. C. Greubel, Jul 15 2019
(GAP) List([0..30], n-> Fibonacci(4*n)+1); # G. C. Greubel, Jul 15 2019

Crossrefs

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

Sequence in context: A368562 A027391 A134988 * A222012 A057834 A196795

Adjacent sequences: A080999 A081000 A081001 * A081003 A081004 A081005

Keyword

nonn,easy

Author

R. K. Guy, Mar 01 2003

Extensions

More terms from James A. Sellers, Mar 01 2003

Status

approved