A042011 Denominators of continued fraction convergents to sqrt(528).

1, 1, 45, 46, 2069, 2115, 95129, 97244, 4373865, 4471109, 201102661, 205573770, 9246348541, 9451922311, 425130930225, 434582852536, 19546776441809, 19981359294345, 898726585392989, 918707944687334, 41321876151635685, 42240584096323019, 1899907576389848521

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 = 44 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 27 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 (0,46,0,-1).

Crossrefs

Cf. A042010, A040505, A002530, A041241.

Programs

Magma

I:=[1,1,45,46]; [n le 4 select I[n] else 46*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 12 2014

Mathematica

Denominator[Convergents[Sqrt[528], 20]] (* Harvey P. Dale, Nov 14 2011 *)
CoefficientList[Series[(1 + x - x^2)/(x^4 - 46 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 12 2014 *)

Formula

G.f.: -(x^2-x-1) / (x^4-46*x^2+1). - Colin Barker, Nov 29 2013

a(n) = 46*a(n-2) - a(n-4) for n > 3. - Vincenzo Librandi, Jan 12 2014

From Peter Bala, May 27 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = sqrt(11) + sqrt(12) and beta = sqrt(11) - sqrt(12) be the roots of the equation x^2 - sqrt(44)*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)} ( 44 + 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) = 44*a(2*n) + a(2*n - 1). (End)

a(2*n) = A041241(2*n) = numerator of continued fraction [4,11,4,11,...,4,11] with n pairs of 4,11. - Greg Dresden, Aug 10 2021

Extensions

More terms from Colin Barker, Nov 29 2013