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A006703
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Solution to Pellian: y such that x^2 - n*y^2 = +-1.
(Formerly M0399)
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8
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0, 1, 1, 0, 1, 2, 3, 1, 0, 1, 3, 2, 5, 4, 1, 0, 1, 4, 39, 2, 12, 42, 5, 1, 0, 1, 5, 24, 13, 2, 273, 3, 4, 6, 1, 0, 1, 6, 4, 3, 5, 2, 531, 30, 24, 3588, 7, 1, 0, 1, 7, 90, 25, 66, 12, 2, 20, 13, 69, 4, 3805, 8, 1, 0, 1, 8, 5967, 4, 936, 30, 413, 2, 125, 5, 3, 6630, 40, 6, 9
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OFFSET
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1,6
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REFERENCES
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A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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MATHEMATICA
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r[x_, n_] := Reduce[ y > 0 && (x^2 - n*y^2 == -1 || x^2 - n*y^2 == 1 ), y, Integers]; a[n_ /; IntegerQ[ Sqrt[n]]] = 0; a[n_] := a[n] = (k = 1; While[r[k, n] === False, k++]; y /. ToRules[r[k, n]]); Table[ Print[ a[n] ]; a[n], {n, 1, 79}] (* Jean-François Alcover, Jan 30 2012 *)
nmax = 500;
nconv = 200; (* The number of convergents'nconv' should be increased if the linear recurrence is not found for some terms. *)
x[n_] := x[n] = Module[{lr}, If[IntegerQ[Sqrt[n]], 1, lr = FindLinearRecurrence[ Numerator[ Convergents[Sqrt[n], nconv]]]; SelectFirst[lr, # > 1 &]/2]];
a[n_] := If[n == 2, 1, SelectFirst[{Sqrt[(x[n]^2 - 1)/n], Sqrt[(x[n]^2 + 1)/n]}, IntegerQ]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Corrected and extended by T. D. Noe, May 19 2007
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STATUS
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approved
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