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A002349
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Take solution to Pellian equation x^2 - n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is a square. A002350 gives values of x.
(Formerly M0046 N0015)
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10
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0, 2, 1, 0, 4, 2, 3, 1, 0, 6, 3, 2, 180, 4, 1, 0, 8, 4, 39, 2, 12, 42, 5, 1, 0, 10, 5, 24, 1820, 2, 273, 3, 4, 6, 1, 0, 12, 6, 4, 3, 320, 2, 531, 30, 24, 3588, 7, 1, 0, 14, 7, 90, 9100, 66, 12, 2, 20, 2574, 69, 4, 226153980, 8, 1, 0, 16, 8, 5967, 4, 936, 30, 413, 2, 267000, 430, 3
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| Albert H. Beiler, "The Pellian" (chap 22), Recreations in the Theory of Numbers, 2nd ed. NY: Dover, 1966.
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, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. E. Whitford, The Pell Equation.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
L. Euler, De solutione problematum diophanteorum per numeros integros, par. 17
E. E. Whitford, The Pell equation, New York, 1912.
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EXAMPLE
| For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 0), (9, 4).
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MATHEMATICA
| a[n_] := If[IntegerQ[Sqrt[n]], 0, For[y=1, !IntegerQ[Sqrt[n*y^2+1]], y++, Null]; y]
PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cof = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cof]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[2]], 0]; Table[ f[n], {n, 0, 75}]
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CROSSREFS
| Cf. A002350, A006702, A006703, A006704, A006705. See A033316, A033315, A033319 for records.
Sequence in context: A062173 A004558 A129699 * A096794 A106375 A194734
Adjacent sequences: A002346 A002347 A002348 * A002350 A002351 A002352
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Enoch Haga (Enokh(AT)comcast.net), Mar 14 2002. Better description from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 14 2003
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