A041022 Numerators of continued fraction convergents to sqrt(15).

3, 4, 27, 31, 213, 244, 1677, 1921, 13203, 15124, 103947, 119071, 818373, 937444, 6443037, 7380481, 50725923, 58106404, 399364347, 457470751, 3144188853, 3601659604, 24754146477, 28355806081

Offset

0,1

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,8,0,-1).

Formula

G.f.: (3+4*x+3*x^2-x^3)/(1-8*x^2+x^4).

From Gerry Martens, Jul 11 2015: (Start)

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

a0(n) = (-((4-sqrt(15))^n*(3+sqrt(15)))+(-3+sqrt(15))*(4+sqrt(15))^n)/2.

a1(n) = ((4-sqrt(15))^n+(4+sqrt(15))^n)/2. (End)

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[15], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
Numerator[Convergents[Sqrt[15], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := (-((4-Sqrt[15])^n*(3+Sqrt[15]))+(-3+Sqrt[15])*(4+Sqrt[15])^n)/2 // Simplify
a1[n_] := ((4-Sqrt[15])^n+(4+Sqrt[15])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)

Crossrefs

Cf. A010472, A041023.

Sequence in context: A222112 A032832 A041021 * A157163 A042225 A094084

Adjacent sequences: A041019 A041020 A041021 * A041023 A041024 A041025

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved