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 A060645 a(0) = 0, a(1) = 4, then a(n) = 18*a(n-1) - a(n-2). 10
 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 December 18 08:14 EST 2018. Contains 318219 sequences. (Running on oeis4.)