Search: id:A002350
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%I A002350 M2240 N0890
%S A002350 1,3,2,1,9,5,8,3,1,19,10,7,649,15,4,1,33,17,170,9,55,197,24,5,1,51,26,
%T A002350 127,9801,11,1520,17,23,35,6,1,73,37,25,19,2049,13,3482,199,161,24335,
%U A002350 48,7,1,99,50,649,66249,485,89,15,151,19603,530,31,1766319049,63,8,1
%N A002350 Take solution to Pellian equation x^2 - n*y^2 = 1 with smallest positive 
               y and x >= 0; sequence gives a(n) = x, or 1 if n is a square. A002349 
               gives values of y.
%D A002350 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002350 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002350 L. Beeckmans, Squares expressible as sum of consecutive squares, Amer. 
               Math. Monthly, 101 (1994), 437-442.
%D A002350 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 A002350 C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
%D A002350 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 
               105, National Research Council, Washington, DC, 1941, p. 55.
%H A002350 T. D. Noe, <a href="b002350.txt">Table of n, a(n) for n=1..1000</a>
%H A002350 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E029.html">
               De solutione problematum diophanteorum per numeros integros</a>, 
               par. 17
%e A002350 For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 
               0), (9, 4).
%t A002350 PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ 
               Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ 
               ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; 
               f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[1]], 1]; Table[ f[n], 
               {n, 0, 65}]
%Y A002350 Cf. A002349, A006702, A006703, A006704, A006705. See A033316, A033315, 
               A033319 for records.
%Y A002350 Sequence in context: A156647 A160760 A152860 this_sequence A109267 A155788 
               A108073
%Y A002350 Adjacent sequences: A002347 A002348 A002349 this_sequence A002351 A002352 
               A002353
%K A002350 nonn,nice,easy
%O A002350 1,2
%A A002350 N. J. A. Sloane (njas(AT)research.att.com).

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