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A033314 Least D in the Pellian x^2 - D*y^2 = 1 for which x has least solution n. 8
3, 2, 15, 6, 35, 12, 7, 5, 11, 30, 143, 42, 195, 14, 255, 18, 323, 10, 399, 110, 483, 33, 23, 39, 27, 182, 87, 210, 899, 60, 1023, 17, 1155, 34, 1295, 38, 1443, 95, 1599, 105, 1763, 462, 215, 506, 235, 138, 47, 96, 51, 26, 2703, 78, 2915, 21, 3135, 203, 3363 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

The i-th solution pair V(i) = [x(i), y(i)] to the Pellian x^2 - D*y^2 = 1 for a given least solution x(1) = n may be generated through the recurrence V(i+2) = 2*n*V(i+1) - V(i) taking V(0) = [1, 0] and V(1) = [n, sqrt((n^2-1)/a(n))]. V(i) stands for the numerator and denominator of the 2i-th convergent of the continued fraction expansion of sqrt(D).

Thus setting n = 3, for instance, we have D = a(3) = 2 and V(1) = [3, 2] so that along with V(0) = [1, 0] recurrence V(i+2) = 6*V(i+1) - V(i) generates [A001333(2k), A000129(2k)]. Similarly, setting n = 9 generates [A023039, A060645], respectively the numerator and denominator of the 2i-th convergent of sqrt(a(9)), i.e., sqrt(5). - Lekraj Beedassy, Feb 26 2002

LINKS

Ray Chandler, Table of n, a(n) for n = 2..1001

Eric Weisstein's World of Mathematics, Pell Equation.

MATHEMATICA

squarefreepart[n_] :=

  Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]);

pellminx[d_] := Module[{q, p, z}, {q, p} = ContinuedFraction[Sqrt[d]];

  If[OddQ[p // Length], p = Join[p, p]];

  z = FromContinuedFraction[Join[{q}, Drop[p, -1]]]; Numerator[z]]

NMAX = 60; a = {};

For[n = 2, n <= NMAX, n++, s = squarefreepart[n^2 - 1];

sd = s Divisors[Sqrt[(n^2 - 1)/s]]^2;

t = Sort[Transpose[{sd, pellminx[#] & /@ sd}]];

AppendTo[a, Select[t, #[[2]] == n &, 1][[1, 1]]]

]; a (* Herbert Kociemba, Jun 05 2022 *)

CROSSREFS

Cf. A000037, A033313, A033318.

Sequence in context: A332215 A086485 A068310 * A070260 A142705 A072346

Adjacent sequences:  A033311 A033312 A033313 * A033315 A033316 A033317

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified August 10 07:57 EDT 2022. Contains 356036 sequences. (Running on oeis4.)