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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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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*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
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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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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