A042859 Denominators of continued fraction convergents to sqrt(960).

1, 1, 61, 62, 3781, 3843, 234361, 238204, 14526601, 14764805, 900414901, 915179706, 55811197261, 56726376967, 3459393815281, 3516120192248, 214426605350161, 217942725542409, 13290990137894701, 13508932863437110, 823826961944121301, 837335894807558411

Offset: 0

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• frac
• easy

Comments

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 60 and Q = -1. This is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 26 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Lehmer Number
• Index entries for linear recurrences with constant coefficients, signature (0,62,0,-1).

Crossrefs

Cf. A042858, A040929.

Cf. A002530.

Programs

Magma

I:=[1,1,61,62]; [n le 4 select I[n] else 62*Self(n-2)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Dec 25 2013

Mathematica

Denominator[Convergents[Sqrt[960], 30]] (* Vincenzo Librandi, Dec 25 2013 *)

Formula

G.f.: -(x^2-x-1) / ((x^2-8*x+1)*(x^2+8*x+1)). - Colin Barker, Dec 25 2013

a(n) = 62*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Dec 25 2013

From Peter Bala, May 26 2014: (Start)

The following remarks assume an offset of 1:

Let alpha = sqrt(15) + 4 and beta = sqrt(15) - 4 be the roots of the equation x^2 - sqrt(60)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.

a(n) = product {k = 1..floor((n-1)/2)} ( 60 + 4*cos^2(k*Pi/n) ). Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 60*a(2*n) + a(2*n - 1). (End)

Extensions

More terms from Colin Barker, Dec 25 2013