Search: id:A060645
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%I A060645
%S A060645 0,4,72,1292,23184,416020,7465176,133957148,2403763488,43133785636,
%T A060645 774004377960,13888945017644,249227005939632,4472197161895732,
%U A060645 80250321908183544,1440033597185408060,25840354427429161536
%N A060645 a(0) = 0, a(1) = 4 then a(n) = 18*a(n-1)-a(n-2).
%C A060645 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].
%C A060645 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.
%C A060645 n such that 5*n^2=floor(sqrt(5)*n*ceil(sqrt(5)*n)) Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 10 2003
%H A060645 Harry J. Smith, <a href="b060645.txt">Table of n, a(n) for n=0,...,200</
               a>
%H A060645 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A060645 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A060645 Author?, <a href="http://www.chemistrycoach.com/nature_of_mathematics_and_mathem.htm">
               Title?</a>
%H A060645 John Robertson, <a href="http://www.jpr2718.org/">Home page.</a>
%F A060645 G.f.: 4x/(1-18*x+x^2). - Cino Hilliard (hillcino368(AT)gmail.com), Feb 
               02 2006
%F A060645 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.
%F A060645 n such that Mod(sigma(5*n^2+1), 2 ) = 1 - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), 
               Mar 26 2004
%F A060645 a(n)=4*A049660(n), a(n)=A000045(6*n)/2. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Feb 03 2006
%F A060645 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
%F A060645 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]
%e A060645 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.
%p A060645 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:
%t A060645 CoefficientList[ Series[4x/(1 - 18x + x^2), {x, 0, 16}], x] (* Robert 
               G. Wilson v *)
%o A060645 (PARI) g(n,k) = for(y=0,n,x=k*y^2+1;if(issquare(x),print1(y",")))
%o A060645 (PARI) a(n)=fibonacci(6*n)/2 (Cloitre)
%o A060645 (PARI) for (i=1,10000,if(Mod(sigma(5*i^2+1),2)==1,print1(i,",")))
%o A060645 (PARI) { for (n=0, 200, write("b060645.txt", n, " ", fibonacci(6*n)/2); 
               ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 09 2009]
%Y A060645 Cf. A023039.
%Y A060645 Sequence in context: A165212 A100521 A111868 this_sequence A003718 A012947 
               A013066
%Y A060645 Adjacent sequences: A060642 A060643 A060644 this_sequence A060646 A060647 
               A060648
%K A060645 nonn
%O A060645 0,2
%A A060645 Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 17 2001
%E A060645 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 19 2001
%E A060645 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Aug 13 2006

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