A042623 Denominators of continued fraction convergents to sqrt(840).

1, 1, 57, 58, 3305, 3363, 191633, 194996, 11111409, 11306405, 644270089, 655576494, 37356553753, 38012130247, 2166035847585, 2204047977832, 125592722606177, 127796770584009, 7282211875310681, 7410008645894690, 422242696045413321

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 = 56 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 W. Weisstein, MathWorld: Lehmer Number

Index entries for linear recurrences with constant coefficients, signature (0, 58, 0, -1).

Formula

G.f.: -(x^2-x-1) / (x^4-58*x^2+1). - Colin Barker, Dec 20 2013

From Peter Bala, May 27 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = sqrt(14) + sqrt(15) and beta = sqrt(14) - sqrt(15) be the roots of the equation x^2 - sqrt(56)*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)} ( 56 + 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) = 56*a(2*n) + a(2*n - 1). (End)

Mathematica

Denominator[Convergents[Sqrt[840], 30]] (* Vincenzo Librandi, Jan 26 2014 *)
LinearRecurrence[{0, 58, 0, -1}, {1, 1, 57, 58}, 30] (* Harvey P. Dale, Oct 31 2016 *)

Crossrefs

Cf. A042622, A040811. A002530.

Sequence in context: A217923 A346806 A136542 * A072466 A354170 A345504

Adjacent sequences: A042620 A042621 A042622 * A042624 A042625 A042626

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

More terms from Colin Barker, Dec 20 2013

Status

approved