A081078 a(n) = Lucas(4n) - 3, or Lucas(2n-1)*Lucas(2n+1).

4, 44, 319, 2204, 15124, 103679, 710644, 4870844, 33385279, 228826124, 1568397604, 10749957119, 73681302244, 505019158604, 3461452807999, 23725150497404, 162614600673844, 1114577054219519, 7639424778862804, 52361396397820124, 358890350005878079

Offset: 1

Author

R. K. Guy, Mar 04 2003

Keywords

• nonn
• easy

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1196
• Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Crossrefs

Cf. A000032 (Lucas numbers), A056854 (Lucas(4n)), A244847.

Programs

Maple

luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 1 to 40 do printf(`%d,`,luc(4*n)-3) od: # James Sellers, Mar 05 2003

Mathematica

LinearRecurrence[{8,-8,1},{4,44,319},30] (* Harvey P. Dale, Jun 08 2014 *)
Table[LucasL[4*n] - 3, {n, 1, 20}] (* Amiram Eldar, Oct 27 2020 *)

PARI

Vec(x*(-4-12*x+x^2) / ( (x-1)*(x^2-7*x+1)) + O(x^30)) \\ Michel Marcus, Dec 23 2014

Formula

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

From R. J. Mathar, Sep 03 2010: (Start)

G.f.: x*(-4-12*x+x^2) / ( (x-1)*(x^2-7*x+1)).

a(n) = A056854(n)-3. (End)

From Peter Bala, Nov 30 2013: (Start)

a(n) = Lucas(2*n)^2 - 5.

Sum_{n>=1} 1/a(n) = (5 - sqrt(5))/10. (End)

Sum_{n>=1} 1/a(n) = A244847. - Amiram Eldar, Oct 27 2020