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A060645
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a(0) = 0, a(1) = 4 then a(n) = 18*a(n-1)-a(n-2).
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6
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0, 4, 72, 1292, 23184, 416020, 7465176, 133957148, 2403763488, 43133785636, 774004377960, 13888945017644, 249227005939632, 4472197161895732, 80250321908183544, 1440033597185408060, 25840354427429161536
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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].
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
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LINKS
| Harry J. Smith, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
John Robertson, Home page.
Index to sequences with linear recurrences with constant coefficients, signature (18,-1).
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FORMULA
| 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 (paoloplava(AT)gmail.com), Oct 02 2008]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), 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)
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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.
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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:
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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] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 10 2010]
LinearRecurrence[{18, -1} {0, 4}, 50] (* Sture Sjöstedt Nov 29 2011 *)
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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 (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]
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CROSSREFS
| Cf. A023039.
Sequence in context: A165212 A100521 A111868 * A203073 A201976 A176901
Adjacent sequences: A060642 A060643 A060644 * A060646 A060647 A060648
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KEYWORD
| nonn,easy
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AUTHOR
| Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 17 2001
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EXTENSIONS
| 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
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