A033317 Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.

2, 1, 4, 2, 3, 1, 6, 3, 2, 180, 4, 1, 8, 4, 39, 2, 12, 42, 5, 1, 10, 5, 24, 1820, 2, 273, 3, 4, 6, 1, 12, 6, 4, 3, 320, 2, 531, 30, 24, 3588, 7, 1, 14, 7, 90, 9100, 66, 12, 2, 20, 2574, 69, 4, 226153980, 8, 1, 16, 8, 5967, 4, 936, 30, 413, 2, 267000, 430, 3, 6630, 40, 6, 9

Offset

1,1

Comments

D = D(n) = A000037(n). - Wolfdieter Lang, Oct 04 2015

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Laurent Beeckmans, Squares Expressible as Sum of Consecutive Squares, Am. Math. Monthly, Volume 101, Number 5, page 442, May 1994.

S. R. Finch, Class number theory [Cached copy, with permission of the author]

Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris. See column B page 19.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Eric Weisstein's World of Mathematics, Pell Equation

Formula

a(n) = sqrt((A033313(n)^2 - 1)/A000037(n)). - Jinyuan Wang, Jul 09 2020

Maple

F:= proc(d) local r, Q; uses numtheory;
  Q:= cfrac(sqrt(d), 'periodic', 'quotients'):
  r:= nops(Q[2]);
  if r::odd then
    denom(cfrac([op(Q[1]), op(Q[2]), op(Q[2][1..-2])]))
  else
    denom(cfrac([op(Q[1]), op(Q[2][1..-2])]));
  fi
end proc:
map(F, remove(issqr, [$1..100])); # Robert Israel, May 17 2015

Mathematica

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2n]; s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
A033317 = DeleteCases[PellSolve /@ Range[100], {}][[All, 2]] (* Jean-François Alcover, Nov 21 2020, after N. J. A. Sloane in A002349 *)

Crossrefs

Cf. A000037, A033313 (for the x's), A077232, A077233.

Sequence in context: A265911 A363893 A078458 * A183200 A326732 A305422

Adjacent sequences: A033314 A033315 A033316 * A033318 A033319 A033320

Keyword

nonn

Author

Eric W. Weisstein

Status

approved