A042739 Denominators of continued fraction convergents to sqrt(899).

1, 1, 59, 60, 3539, 3599, 212281, 215880, 12733321, 12949201, 763786979, 776736180, 45814485419, 46591221599, 2748105338161, 2794696559760, 164840505804241, 167635202364001, 9887682242916299, 10055317445280300, 593096094069173699

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 = 58 and Q = -1. This 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 26 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 (0,60,0,-1).

Formula

G.f.: -(x^2-x-1) / (x^4-60*x^2+1). - Colin Barker, Dec 22 2013

From Peter Bala, May 26 2014: (Start)

The following remarks assume an offset of 1. Let alpha = ( sqrt(58) + sqrt(62) )/2 and beta = ( sqrt(58) - sqrt(62) )/2 be the roots of the equation x^2 - sqrt(58)*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)} (58 + 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) = 58*a(2*n) + a(2*n - 1). (End)

Mathematica

Denominator[Convergents[Sqrt[899], 30]] (* Vincenzo Librandi, Jan 28 2014 *)
LinearRecurrence[{0, 60, 0, -1}, {1, 1, 59, 60}, 30] (* Harvey P. Dale, Apr 01 2017 *)

Crossrefs

Cf. A042738, A040869. A002530.

Sequence in context: A292093 A104916 A172462 * A348570 A172256 A172056

Adjacent sequences: A042736 A042737 A042738 * A042740 A042741 A042742

Keyword

nonn,frac,easy

Author

N. J. A. Sloane.

Extensions

Additional term from Colin Barker, Dec 22 2013

Status

approved