A081010 a(n) = Fibonacci(4n+1) + 2, or Fibonacci(2n-1)*Lucas(2n+2).

3, 7, 36, 235, 1599, 10948, 75027, 514231, 3524580, 24157819, 165580143, 1134903172, 7778742051, 53316291175, 365435296164, 2504730781963, 17167680177567, 117669030460996, 806515533049395, 5527939700884759, 37889062373143908, 259695496911122587

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), A001519, A001906, A322159.

Programs

GAP

List([0..30], n-> Fibonacci(4*n+1)+2); # G. C. Greubel, Jul 14 2019

Magma

[Fibonacci(4*n+1) +2: 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)+2) od # James Sellers, Mar 03 2003

Mathematica

Fibonacci[4*Range[0,30]+1]+2 (* G. C. Greubel, Jul 14 2019 *)

PARI

vector(30, n, n--; fibonacci(4*n+1)+2) \\ G. C. Greubel, Jul 14 2019

Sage

[fibonacci(4*n+1)+2 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).

a(n) = 2 + (A001906(n+1)^2 + A001519(n)^2)/2. - Creighton Dement, Aug 15 2004

G.f.: (3-17*x+4*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) = A322159. - Amiram Eldar, Nov 28 2024

Extensions

More terms from James Sellers, Mar 03 2003