A041062 Numerators of continued fraction convergents to sqrt(38).

6, 37, 450, 2737, 33294, 202501, 2463306, 14982337, 182251350, 1108490437, 13484136594, 82013310001, 997643856606, 6067876449637, 73812161252250, 448940843963137, 5461102288809894, 33215554576822501

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Index entries for linear recurrences with constant coefficients, signature (0,74,0,-1).

Formula

G.f.: -(x^3-6*x^2-37*x-6) / (x^4-74*x^2+1). - Colin Barker, Nov 04 2013

From Gerry Martens, Jul 11 2015: (start)

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

a0(n) = (-3+sqrt(19/2))*(37+6*sqrt(38))^n-(6+sqrt(38))/(2*(37+6*sqrt(38))^n).

a1(n) = (1/(37+6*sqrt(38))^n+(37+6*sqrt(38))^n)/2. (End)

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[38], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)
Numerator[Convergents[Sqrt[38], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
a0[n_] := (-3+Sqrt[19/2])*(37+6*Sqrt[38])^n-(6+Sqrt[38])/(2*(37+6*Sqrt[38])^n) // Simplify
a1[n_] := (1/(37+6*Sqrt[38])^n+(37+6*Sqrt[38])^n)/2 // FullSimplify
Flatten[MapIndexed[{a0[#], a1[#]}&, Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0, 74, 0, -1}, {6, 37, 450, 2737}, 20] (* Harvey P. Dale, Oct 17 2020 *)

Crossrefs

Cf. A041063, A010492.

Sequence in context: A340029 A185036 A185236 * A240324 A283636 A211988

Adjacent sequences: A041059 A041060 A041061 * A041063 A041064 A041065

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved