A041059 Denominators of continued fraction convergents to sqrt(35).

1, 1, 11, 12, 131, 143, 1561, 1704, 18601, 20305, 221651, 241956, 2641211, 2883167, 31472881, 34356048, 375033361, 409389409, 4468927451, 4878316860, 53252096051, 58130412911, 634556225161, 692686638072, 7561422605881, 8254109243953, 90102515045411

Offset: 0

Author

N. J. A. Sloane

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 = 10 and Q = -1; it 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 28 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 (0,12,0,-1).

Crossrefs

Cf. A010490, A041058, A002539.

Programs

Mathematica

Denominator[Convergents[Sqrt[35], 30]] (* Vincenzo Librandi, Oct 23 2013 *)

Formula

G.f.: (1+x-x^2)/(1-12*x^2+x^4). - Colin Barker, Jan 01 2012

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(10) + sqrt(14) )/2 and beta = ( sqrt(10) - sqrt(14) )/2 be the roots of the equation x^2 - sqrt(10)*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)} ( 10 + 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) = 10*a(2*n) + a(2*n - 1). (End)