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A081014 a(n) = Lucas(4*n+1) + 1, or Lucas(2*n)*Lucas(2*n+1). 1
2, 12, 77, 522, 3572, 24477, 167762, 1149852, 7881197, 54018522, 370248452, 2537720637, 17393796002, 119218851372, 817138163597, 5600748293802, 38388099893012, 263115950957277, 1803423556807922, 12360848946698172 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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).

FORMULA

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

a(n) = 1 + (1/2)*(((7/2)-(3/2)*sqrt(5))^n + ((7/2)+(3/2)*sqrt(5))^n + (1/2)*sqrt(5)*(((7/2)+(3/2)*sqrt(5))^n  - ((7/2)-(3/2)*sqrt(5))^n), with n >= 0. - Paolo P. Lava, Dec 01 2008

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

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 A. 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 *)

PROG

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

(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

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

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

CROSSREFS

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

Sequence in context: A285489 A121680 A277478 * A223771 A062871 A306272

Adjacent sequences:  A081011 A081012 A081013 * A081015 A081016 A081017

KEYWORD

nonn,easy

AUTHOR

R. K. Guy, Mar 01 2003

STATUS

approved

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Last modified December 5 12:24 EST 2021. Contains 349557 sequences. (Running on oeis4.)