A042511 Denominators of continued fraction convergents to sqrt(783).

1, 1, 55, 56, 3079, 3135, 172369, 175504, 9649585, 9825089, 540204391, 550029480, 30241796311, 30791825791, 1693000389025, 1723792214816, 94777779989089, 96501572203905, 5305862678999959, 5402364251203864, 297033532244008615, 302435896495212479

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 = 54 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 (0,56,0,-1).

Formula

G.f.: -(x^2-x-1) / (x^4-56*x^2+1). - Colin Barker, Dec 16 2013

From Peter Bala, May 27 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(54) + sqrt(58) )/2 and beta = ( sqrt(54) - sqrt(58) )/2 be the roots of the equation x^2 - sqrt(54)*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)} ( 54 + 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) = 54*a(2*n) + a(2*n - 1). (End)

Mathematica

Denominator[Convergents[Sqrt[783], 30]] (* Vincenzo Librandi, Jan 23 2014 *)

Crossrefs

Cf. A042510, A040755. A002530.

Sequence in context: A112892 A232653 A291502 * A020282 A345483 A345484

Adjacent sequences: A042508 A042509 A042510 * A042512 A042513 A042514

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

More terms from Colin Barker, Dec 16 2013

Status

approved