0,2

a(n) = 18*a(n-1) - a(n-2), with a(1) = denominator of continued fraction [2;4] and a(2) = denominator of [2;4,4,4].

a(n) solves for y in the Diophantine equation x^2 - 5*y^2 = 1, the third simplest case of the Pell-Fermat type. The corresponding x solutions are provided by A023039.

n such that 5*n^2=floor(sqrt(5)*n*ceil(sqrt(5)*n)) Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2003

Harry J. Smith, Table of n, a(n) for n=0,...,200

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

Author?, Title?

John Robertson, Home page.

G.f.: 4x/(1-18*x+x^2). - Cino Hilliard (hillcino368(AT)gmail.com), Feb 02 2006

a(n) may be computed either as i) the denominator of the (2n-1)-th convergent of the continued fraction [2;4, 4, 4, ...] = sqrt(5), or ii) as the coefficient of sqrt(5) in {9+sqrt(5)}^n.

n such that Mod(sigma(5*n^2+1), 2 ) = 1 - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004

a(n)=4*A049660(n), a(n)=A000045(6*n)/2. - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 03 2006

a(n) = 17*(a(n-1)+a(n-2))-a(n-3), a(n) = 19*(a(n-1)-a(n-2))+a(n-3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 20 2006

a(n)=-(1/10)*[9-4*sqrt(5)]^n*sqrt(5)+(1/10)*sqrt(5)*[9+4*sqrt(5)]^n, with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Oct 02 2008]

Given a(1) = 4, a(2) = 72 we have for instance a(4) = 18*a(3) - a(2) = 18*{18*a(2) -a(1)} - a(2), i.e. a(4) = 323*a(2) - 18*a(1) = 323*72 - 18*4 = 23184.

A060645 := proc(n) option remember: if n=1 then RETURN(4) fi: if n=2 then RETURN(72) fi: 18*A060645(n -1)- A060645(n-2): end: for n from 1 to 30 do printf(`%d, `, A060645(n)) od:

CoefficientList[ Series[4x/(1 - 18x + x^2), {x, 0, 16}], x] (* Robert G. Wilson v *)

(PARI) g(n, k) = for(y=0, n, x=k*y^2+1; if(issquare(x), print1(y", ")))

(PARI) a(n)=fibonacci(6*n)/2 (Cloitre)

(PARI) for (i=1, 10000, if(Mod(sigma(5*i^2+1), 2)==1, print1(i, ", ")))

(PARI) { for (n=0, 200, write("b060645.txt", n, " ", fibonacci(6*n)/2); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 09 2009]

Cf. A023039.

Sequence in context: A165212 A100521 A111868 this_sequence A003718 A012947 A013066

Adjacent sequences: A060642 A060643 A060644 this_sequence A060646 A060647 A060648

nonn

Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 17 2001

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 19 2001

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Aug 13 2006