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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

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).

Crossrefs

Cf. A010469, A041017.

Programs

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 *)

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)