Search: id:A002349
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%I A002349 M0046 N0015
%S A002349 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,
%T A002349 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,
%U A002349 2,20,2574,69,4,226153980,8,1,0,16,8,5967,4,936,30,413,2,267000,430,3
%N A002349 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.
%D A002349 Albert H. Beiler, "The Pellian" (chap 22), Recreations in the Theory 
               of Numbers, 2nd ed. NY: Dover, 1966.
%D A002349 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.
%D A002349 C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
%D A002349 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 
               105, National Research Council, Washington, DC, 1941, p. 55.
%D A002349 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002349 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002349 E. E. Whitford, The Pell Equation.
%H A002349 T. D. Noe, <a href="b002349.txt">Table of n, a(n) for n=1..1000</a>
%H A002349 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E029.html">
               De solutione problematum diophanteorum per numeros integros</a>, 
               par. 17
%H A002349 E. E. Whitford, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;
               c=umhistmath;idno=ABV2773.0001.001">The Pell equation</a>, New York, 
               1912.
%e A002349 For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 
               0), (9, 4).
%t A002349 a[n_] := If[IntegerQ[Sqrt[n]], 0, For[y=1, !IntegerQ[Sqrt[n*y^2+1]], 
               y++, Null]; y]
%t A002349 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}]
%Y A002349 Cf. A002350, A006702, A006703, A006704, A006705. See A033316, A033315, 
               A033319 for records.
%Y A002349 Sequence in context: A062173 A004558 A129699 this_sequence A096794 A106375 
               A131667
%Y A002349 Adjacent sequences: A002346 A002347 A002348 this_sequence A002350 A002351 
               A002352
%K A002349 nonn,nice,easy
%O A002349 1,2
%A A002349 N. J. A. Sloane (njas(AT)research.att.com).
%E A002349 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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