A041449 Denominators of continued fraction convergents to sqrt(240).

1, 2, 61, 124, 3781, 7686, 234361, 476408, 14526601, 29529610, 900414901, 1830359412, 55811197261, 113452753934, 3459393815281, 7032240384496, 214426605350161, 435885451084818, 13290990137894701, 27017865726874220, 823826961944121301, 1674671789615116822

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 (0,62,0,-1).

Crossrefs

Cf. A041448, A040224.

Programs

Magma

I:=[1,2,61,124]; [n le 4 select I[n] else 62*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 18 2013

Mathematica

Denominator[Convergents[Sqrt[240], 30]] (* Vincenzo Librandi, Dec 18 2013 *)
a0[n_] := Sqrt[2+(31-8*Sqrt[15])^(1+2*n)+(31+8*Sqrt[15])^(1+2*n)]/8 // Simplify
a1[n_] := 2*Sum[a0[i], {i, 0, n}]
Flatten[MapIndexed[{a0[#-1],a1[#-1]}&,Range[11]]] (* Gerry Martens, Jul 10 2015 *)

Formula

G.f.: -(x^2-2*x-1) / ((x^2-8*x+1)*(x^2+8*x+1)). - Colin Barker, Nov 17 2013

a(n) = 62*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Dec 18 2013

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n-1),a1(n-1)] for n>0:

a0(n) = sqrt(2+(31-8*sqrt(15))^(2*n+1)+(31+8*sqrt(15))^(2*n+1))/8.

a1(n) = 2*sum(i=0,n,a0(i)). (End)

Extensions

More terms from Colin Barker, Nov 17 2013