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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

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).

Crossrefs

Cf. A041126, A040063, A020829, A010524.

Programs

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

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 *)

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

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)