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A060645 a(0) = 0, a(1) = 4, then a(n) = 18*a(n-1) - a(n-2). 8
0, 4, 72, 1292, 23184, 416020, 7465176, 133957148, 2403763488, 43133785636, 774004377960, 13888945017644, 249227005939632, 4472197161895732, 80250321908183544, 1440033597185408060, 25840354427429161536 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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].

This sequence gives the values of y in solutions of the Diophantine equation x^2 - 5*y^2 = 1, the third simplest case of the Pell-Fermat type. The corresponding x values are in A023039.

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

LINKS

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

Tanya Khovanova, Recursive Sequences

John Robertson, Home page.

Index entries for linear recurrences with constant coefficients, signature (18,-1).

FORMULA

G.f.: 4x/(1-18*x+x^2). - Cino Hilliard, 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, 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. - Paolo P. Lava, Oct 02 2008

From Johannes W. Meijer, Jul 01 2010: (Start)

Limit(a(n+k)/a(k), k=infinity) = A023039(n) + A060645(n)*sqrt(5).

Limit(A023039(n)/a(n), n=infinity) = sqrt(5). (End)

a(n)= fibonacci(6*n)/2. - Gary Detlefs, Apr 02 2012

a(n) = 4*S(n-1, 18), with Chebyshev's S-polynomials. See A049310. S(-1, x)= 0. - Wolfdieter Lang, Aug 24 2014

EXAMPLE

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.

MAPLE

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:

MATHEMATICA

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

Select[Select[Table[Fibonacci[n], {n, 0, 5!}], EvenQ]/2, EvenQ] (* Vladimir Joseph Stephan Orlovsky, May 10 2010 *)

LinearRecurrence[{18, -1} {0, 4}, 50] (* Sture Sjöstedt, Nov 29 2011 *)

PROG

(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 \\ Benoit 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); ) } \\ Harry J. Smith, Jul 09 2009

CROSSREFS

Cf. A023039.

Sequence in context: A263219 A100521 A111868 * A203073 A231033 A201976

Adjacent sequences:  A060642 A060643 A060644 * A060646 A060647 A060648

KEYWORD

nonn,easy

AUTHOR

Lekraj Beedassy, Apr 17 2001

EXTENSIONS

More terms from James A. Sellers, Apr 19 2001

Entry revised by N. J. A. Sloane, Aug 13 2006

STATUS

approved

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Last modified June 24 05:50 EDT 2017. Contains 288697 sequences.