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Search: id:A075796
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| A075796 |
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Numbers k such that 5*k^2 + 5 is a square. |
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+0
2
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| 2, 38, 682, 12238, 219602, 3940598, 70711162, 1268860318, 22768774562, 408569081798, 7331474697802, 131557975478638, 2360712083917682, 42361259535039638, 760141959546795802, 13640194012307284798
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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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Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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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).
a(n+1)=9*a(n)+4*sqrt(5)*(a(n)^2+1)^0.5. - Richard Choulet (richardchoulet(AT)yahoo.fr), Aug 30 2007
G.f.: f(z)=a(1)*z+a(2)*z^2+...=(2*z+2*z^2)/(1-18*z+z^2) - R. Choulet (richardchoulet(AT)yahoo.fr), Oct 09 2007
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CROSSREFS
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Sequence in context: A126731 A046845 A098772 this_sequence A132396 A066244 A055689
Adjacent sequences: A075793 A075794 A075795 this_sequence 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 (omomom(AT)hotmail.com), Oct 13 2002
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