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A006704
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Solution to Pellian: x such that x^2 - n y^2 = +- 1, +- 4.
(Formerly M0119)
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5
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1, 1, 2, 1, 1, 5, 8, 2, 1, 3, 10, 4, 3, 15, 4, 1, 4, 17, 170, 4, 5, 197, 24, 5, 1, 5, 26, 16, 5, 11, 1520, 6, 23, 35, 6, 1, 6, 37, 25, 6, 32, 13, 3482, 20, 7, 24335, 48, 7, 1, 7, 50, 36, 7, 485, 89, 15, 151, 99, 530, 8, 39, 63, 8, 1, 8, 65, 48842, 8, 25, 251, 3480, 17, 1068, 43
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OFFSET
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1,3
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COMMENTS
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The definition of the sequence is unclear. - Michael Somos, Mar 07 2012
When n is a square, the trivial solution (x,y) = (1,0) is taken; otherwise we take the least nontrivial solution that satisfies one of the four equations with +1, -1, +4 or -4. - Ray Chandler, Aug 22 2015
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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[lhs = x^2 - n*y^2; y > 0 && (lhs == -1 || lhs == 1 || lhs == -4 || lhs == 4), y, Integers]; a[n_ /; IntegerQ[Sqrt[n]]] = 1; a[n_] := (x = 1; While[r[x, n] === False, x++]; x); yy[1, _] = 1; yy[x_, n_] := r[x, n][[2]]; A006704 = Table[x = a[n]; Print[{n, x, yy[x, n]}]; x, {n, 1, 55}] (* Jean-François Alcover, Mar 07 2012 *)
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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 a(47)=48 and extended by Ray Chandler, Aug 22 2015
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STATUS
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approved
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