

A006702


Solution to a Pellian equation: least x such that x^2  n*y^2 = + 1.
(Formerly M0120)


10



1, 1, 2, 1, 2, 5, 8, 3, 1, 3, 10, 7, 18, 15, 4, 1, 4, 17, 170, 9, 55, 197, 24, 5, 1, 5, 26, 127, 70, 11, 1520, 17, 23, 35, 6, 1, 6, 37, 25, 19, 32, 13, 3482, 199, 161, 24335, 48, 7, 1, 7, 50, 649, 182, 485, 89, 15, 151, 99, 530, 31, 29718, 63, 8, 1, 8, 65, 48842
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OFFSET

1,3


COMMENTS

When n is a square, the trivial solution x=1, y=0 is taken; otherwise we take the least x that satisfies either the +1 or 1 equation.  T. D. Noe, May 19 2007
Apparently the generating function of the sequence of the denominators of continued fraction convergents to sqrt(n) is always rational and of form p(x)/[1  C*x^m + (1)^m * x^(2m)], or equivalently, the denominators satisfy the linear recurrence b(n+2m) = C*b(n+m)  (1)^m * b(n). If so, then it seems that a(n) is half the value of C for each nonsquare n, or 1. See A003285 for the conjecture regarding m. The same conjectures apply to the sequences of the numerators of continued fraction convergents to sqrt(n).  Ralf Stephan, Dec 12 2013


REFERENCES

A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 113, Cambridge Univ. Press, London, 18891897, Vol. 13, pp. 430443.
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).


LINKS



MATHEMATICA

r[x_, n_] := Reduce[y > 0 && (x^2  n*y^2 == 1  x^2  n*y^2 == 1 ), y, Integers];
a[n_ /; IntegerQ[ Sqrt[n]]] = 1;
a[n_] := a[n] = (k = 1; While[ r[k, n] === False, k++]; k);
nmax = 500;
nconv = 200; (* The number of convergents 'nconv' should be increased if the linear recurrence is not found for some terms. *)
a[n_] := a[n] = Module[{lr}, If[IntegerQ[Sqrt[n]], 1, lr = FindLinearRecurrence[Numerator[Convergents[Sqrt[n], nconv]]]; SelectFirst[lr, #>1&]/2]];


CROSSREFS



KEYWORD

nonn,nice,easy


AUTHOR



EXTENSIONS

Corrected and extended by T. D. Noe, May 19 2007


STATUS

approved



