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A075796
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Numbers k such that 5*k^2 + 5 is a square.
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4
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2, 38, 682, 12238, 219602, 3940598, 70711162, 1268860318, 22768774562, 408569081798, 7331474697802, 131557975478638, 2360712083917682, 42361259535039638, 760141959546795802, 13640194012307284798
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OFFSET
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1,1
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COMMENTS
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Lim. n-> Inf. a(n)/a(n-1) = 8*phi + 1 = 9 + 4*Sqrt(5).
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REFERENCES
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A. H. Beiler, "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..200
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
Index to sequences with linear recurrences with constant coefficients, signature (18,-1).
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FORMULA
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a(n) = [ [(9 + 4*Sqrt(5))^n - (9 - 4*Sqrt(5))^n] + [(9 + 4*Sqrt(5))^(n-1) - (9 - 4*Sqrt(5))^(n-1)] ] / (4*Sqrt(5))
a(n) = 18*a(n-1) - a(n-2). a(n) = 2*A049629(n-1).
a(n+1)=9*a(n)+4*sqrt(5)*(a(n)^2+1)^0.5. - Richard Choulet, Aug 30 2007
G.f.: 2*x*(1+x)/( 1-18*x+x^2 ). - Richard Choulet, Oct 09 2007
Contribution from Johannes W. Meijer, Jul 01 2010: (Start)
a(n) = A000045(6*n+3) + A000045(6*n)/2.
a(n) = 2*A167808(6*n+4) - A167808(6*n+6).
Limit(a(n+k)/a(k), k=infinity) = A023039(n)*A060645(n)*sqrt(5)
(End)
5*A007805(n)^2 -1 = a(n+1)^2. - Sture Sjöstedt, Nov 29 2011
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MATHEMATICA
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LinearReccurrence[{18, -1}, {2, 38}, 50] ( *Sture Sjöstedt, Nov 29 2011*)
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PROG
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(MAGMA) I:=[2, 38]; [n le 2 select I[n] else 18*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 30 2011
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CROSSREFS
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Sequence in context: A098772 A207320 A208240 * A187544 A187545 A184994
Adjacent sequences: A075793 A075794 A075795 * A075797 A075798 A075799
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KEYWORD
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nonn
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AUTHOR
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Gregory V. Richardson, Oct 13 2002
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EXTENSIONS
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Typo in Mathematica program fixed by Vincenzo Librandi, Nov 30 2011
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STATUS
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approved
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