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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

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).

Crossrefs

Cf. A041063, A010492.

Programs

Mathematica

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 *)

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)