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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.
(Formerly M0046 N0015)
21

%I M0046 N0015

%S 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,

%T 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,

%U 12,2,20,2574,69,4,226153980,8,1,0,16,8,5967,4,936,30,413,2,267000,430,3

%N 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 Albert H. Beiler, "The Pellian" (chap 22), Recreations in the Theory of Numbers, 2nd ed. NY: Dover, 1966.

%D 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 C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.

%D D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D E. E. Whitford, The Pell Equation.

%H T. D. Noe and Ray Chandler, <a href="/A002349/b002349.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H A. Cayley, <a href="/A002349/a002349.pdf">Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ...</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443. (Annotated scanned copy)

%H L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E029.html">De solutione problematum diophanteorum per numeros integros</a>, par. 17.

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)

%H 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 For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 0), (9, 4).

%t a[n_] := If[IntegerQ[Sqrt[n]], 0, For[y=1, !IntegerQ[Sqrt[n*y^2+1]], y++, Null]; y]

%t (* Second program: *)

%t PellSolve[(m_Integer)?Positive] := Module[{cof, 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 Cf. A002350, A006702, A006703, A006704, A006705. See A033316, A033315, A033319 for records.

%K nonn,nice,easy,changed

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Enoch Haga_, Mar 14 2002

%E Better description from _Robert G. Wilson v_, Apr 14 2003

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Last modified October 22 08:00 EDT 2019. Contains 328315 sequences. (Running on oeis4.)