A041127 Denominators of continued fraction convergents to sqrt(72).

1, 2, 33, 68, 1121, 2310, 38081, 78472, 1293633, 2665738, 43945441, 90556620, 1492851361, 3076259342, 50713000833, 104502261008, 1722749176961, 3550000614930, 58522759015841, 120595518646612, 1988051057361633, 4096697633369878, 67535213191279681

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Formula

G.f.: -(x^2-2*x-1) / ((x^2-6*x+1)*(x^2+6*x+1)). - Colin Barker, Nov 13 2013

a(n) = 34*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 11 2013

From Gerry Martens, Jul 11 2015: (Start)

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

a0(n) = ((3+2*sqrt(2))/(17+12*sqrt(2))^n+(3-2*sqrt(2))*(17+12*sqrt(2))^n)/6.

a1(n) = (-1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/(12*sqrt(2)). (End)

Mathematica

Denominator/@Convergents[Sqrt[72], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[(1 + 2 x - x^2)/(x^4 - 34 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
a0[n_] := ((3+2*Sqrt[2])/(17+12*Sqrt[2])^n+(3-2*Sqrt[2])*(17+ 12*Sqrt[2])^n)/6 // Simplify
a1[n_] := (-1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/(12*Sqrt[2]) // FullSimplify
Flatten[MapIndexed[{a0[#], a1[#]}&, Range[20]]] (* Gerry Martens, Jul 10 2015 *)

Prog

(PARI) a(n)=my(v=contfrac(sqrt(72), n), s=v[n]); forstep(k=n-1, 1, -1, s=v[k]+1/s); denominator(s) \\ Charles R Greathouse IV, Jul 05 2011
(Magma) I:=[1, 2, 33, 68]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

Crossrefs

Cf. A041126, A040063, A020829, A010524.

Sequence in context: A303375 A342620 A065647 * A282726 A097978 A377353

Adjacent sequences: A041124 A041125 A041126 * A041128 A041129 A041130

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Status

approved