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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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10
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0, 4, 72, 1292, 23184, 416020, 7465176, 133957148, 2403763488, 43133785636, 774004377960, 13888945017644, 249227005939632, 4472197161895732, 80250321908183544, 1440033597185408060, 25840354427429161536
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OFFSET
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0,2
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COMMENTS
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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*ceiling(sqrt(5)*n)). - Benoit Cloitre, May 10 2003
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 0..200
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
John Robertson, Home page.
Index entries for linear recurrences with constant coefficients, signature (18,-1).
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FORMULA
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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].
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 sigma(5*n^2+1) mod 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, 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)
Lim_{k->infinity} a(n+k)/a(k) = A023039(n) + A060645(n)*sqrt(5).
Lim_{n->infinity} A023039(n)/a(n) = 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
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EXAMPLE
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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
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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
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A023039.
Sequence in context: A263219 A100521 A111868 * A203073 A231033 A307358
Adjacent sequences: A060642 A060643 A060644 * A060646 A060647 A060648
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KEYWORD
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nonn,easy
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AUTHOR
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Lekraj Beedassy, Apr 17 2001
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EXTENSIONS
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More terms from James A. Sellers, Apr 19 2001
Entry revised by N. J. A. Sloane, Aug 13 2006
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STATUS
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approved
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