A041016 Numerators of continued fraction convergents to sqrt(12).

3, 7, 45, 97, 627, 1351, 8733, 18817, 121635, 262087, 1694157, 3650401, 23596563, 50843527, 328657725, 708158977, 4577611587, 9863382151, 63757904493, 137379191137, 888033051315, 1913445293767, 12368704813917

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Formula

G.f.: (3+7*x+3*x^2-x^3)/(1-14*x^2+x^4).

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n),a1(n)] for n>0:

a0(n) = (-((7-4*sqrt(3))^n*(3+2*sqrt(3)))+(-3+2*sqrt(3))*(7+4*sqrt(3))^n)/2.

a1(n) = ((7-4*sqrt(3))^n+(7+4*sqrt(3))^n)/2. (End)

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[12], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
Numerator[Convergents[Sqrt[12], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := (-((7-4*Sqrt[3])^n*(3+2*Sqrt[3]))+(-3+2*Sqrt[3])*(7+4*Sqrt[3])^n)/2 //Simplify
a1[n_] := ((7-4*Sqrt[3])^n+(7+4*Sqrt[3])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0, 14, 0, -1}, {3, 7, 45, 97}, 30] (* Harvey P. Dale, Jun 02 2016 *)

Crossrefs

Cf. A010469, A041017.

Sequence in context: A267844 A359046 A041349 * A351354 A365572 A301324

Adjacent sequences: A041013 A041014 A041015 * A041017 A041018 A041019

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved