A081014 a(n) = Lucas(4*n+1) + 1, or Lucas(2*n)*Lucas(2*n+1).

2, 12, 77, 522, 3572, 24477, 167762, 1149852, 7881197, 54018522, 370248452, 2537720637, 17393796002, 119218851372, 817138163597, 5600748293802, 38388099893012, 263115950957277, 1803423556807922, 12360848946698172

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Crossrefs

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

Programs

GAP

List([0..20], n-> Lucas(1,-1,4*n+1)[2] +1); # G. C. Greubel, Jul 14 2019

Magma

I:=[2,12,77]; [n le 3 select I[n] else 8*Self(n-1) - 8*Self(n-2) + Self(n-3): n in [0..30]]; // G. C. Greubel, Dec 24 2017

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 0 to 25 do printf(`%d,`,luc(4*n+1)+1) od: # James Sellers, Mar 03 2003

Mathematica

LinearRecurrence[{8,-8,1}, {2,12,77}, 20] (* G. C. Greubel, Dec 24 2017 *)
LucasL[4*Range[0,20] +1] +1 (* G. C. Greubel, Jul 14 2019 *)
CoefficientList[Series[(2-4x-3x^2)/((1-x)(1-7x+x^2)),{x,0,30}],x] (* Harvey P. Dale, Aug 27 2021 *)

PARI

vector(20, n, n--; f=fibonacci; f(4*n+2)+f(4*n)+1) \\ G. C. Greubel, Dec 24 2017

Sage

[lucas_number2(4*n+1,1,-1) +1 for n in (0..20)] # G. C. Greubel, Jul 14 2019

Formula

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

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

G.f.: (2 -4*x -3*x^2)/((1-x)*(1-7*x+x^2)).

a(n) = 1 + A056914(n). (End)

a(n) = 7*a(n-1) - a(n-2) - 5, n >= 2. - R. J. Mathar, Nov 07 2015