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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.
Numbers k such that 5*k^2 = floor(sqrt(5)*k*ceiling(sqrt(5)*k)). - Benoit Cloitre, May 10 2003
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LINKS
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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].
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.
Numbers k such that sigma(5*k^2 + 1) mod 2 = 1. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004
a(n) = 17*(a(n-1) + a(n-2)) - a(n-3) = 19*(a(n-1) - a(n-2)) + a(n-3). - Mohamed Bouhamida, Sep 20 2006
Limit_{n->infinity} A023039(n)/a(n) = sqrt(5). (End)
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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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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) 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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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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