A041063 Denominators of continued fraction convergents to sqrt(38).

1, 6, 73, 444, 5401, 32850, 399601, 2430456, 29565073, 179820894, 2187415801, 13304315700, 161839204201, 984339540906, 11973913695073, 72827821711344, 885907774231201, 5388274467098550, 65545201379413801, 398659482743581356, 4849458994302390073

Offset

0,2

Links

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

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

Formula

G.f.: -(x^2-6*x-1) / (x^4-74*x^2+1). - Colin Barker, Nov 12 2013

a(n) = 74*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013

From Gerry Martens, Jul 11 2015: (Start)

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

a0(n) = ((38+6*sqrt(38))/(37+6*sqrt(38))^n+(38-6*sqrt(38))*(37+6*sqrt(38))^n)/76;

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

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[38], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)
Denominator[Convergents[Sqrt[38], 30]] (* Vincenzo Librandi, Dec 10 2013 *)
a0[n_] := ((38+6*Sqrt[38])/(37+6*Sqrt[38])^n+(38-6*Sqrt[38])*(37+6*Sqrt[38])^n)/76 // Simplify
a1[n_] := (-1/(37+6*Sqrt[38])^n+(37+6*Sqrt[38])^n)/(2*Sqrt[38]) // FullSimplify
Flatten[MapIndexed[{a0[#], a1[#]}&, Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0, 74, 0, -1}, {1, 6, 73, 444}, 30] (* Harvey P. Dale, Feb 29 2024 *)

Prog

(Magma) I:=[1, 6, 73, 444]; [n le 4 select I[n] else 74*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

Crossrefs

Cf. A010492, A041062.

Sequence in context: A203433 A008562 A367350 * A105324 A179568 A202557

Adjacent sequences: A041060 A041061 A041062 * A041064 A041065 A041066

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Extensions

More terms from Colin Barker, Nov 12 2013

Status

approved