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A075869
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Numbers k such that 5*k^2 - 9 is a square.
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0
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3, 51, 915, 16419, 294627, 5286867, 94868979, 1702354755, 30547516611, 548152944243, 9836205479763, 176503545691491, 3167227616967075, 56833593559715859, 1019837456457918387, 18300240622682815107
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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) = phi^6 = 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, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.
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LINKS
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J. J. O'Connor and E. F. Robertson, Pell's Equation [From the Internet Archive Wayback machine]
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FORMULA
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a(n) = 3*sqrt(5)/10*((2+sqrt(5))^(2*n-1)-(2-sqrt(5))^(2*n-1)) = 18*a(n-1) - a(n-2).
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MATHEMATICA
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LinearRecurrence[{18, -1}, {3, 51}, 20] (* Harvey P. Dale, Dec 27 2018 *)
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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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STATUS
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approved
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