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

0,3

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).

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)

Mathematica

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

Crossrefs

Cf. A010490, A041058, A002539.

Sequence in context: A038326 A249312 A357273 * A041260 A109665 A041261

Adjacent sequences: A041056 A041057 A041058 * A041060 A041061 A041062

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Status

approved