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A006702 Solution to a Pellian equation: least x such that x^2 - n y^2 = +- 1.
(Formerly M0120)
9
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 (list; graph; refs; listen; history; text; internal format)
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 + x^(2m)], or equivalently, the denominators satisfy the linear recurrence b(n+2m) = C*b(n+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. 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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

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. (Annotated scanned copy)

M. Zuker, Fundamental solution to Pell's Equation x^2 - d*y^2 = +-1 [Broken link]

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); Table[ Print[ a[n] ]; a[n], {n, 1, 67}] (* Jean-Fran├žois Alcover, Jan 30 2012 *)

CROSSREFS

Cf. A006703, A077232.

Sequence in context: A153910 A208021 A052532 * A129394 A214899 A199599

Adjacent sequences:  A006699 A006700 A006701 * A006703 A006704 A006705

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified April 26 09:52 EDT 2019. Contains 322472 sequences. (Running on oeis4.)