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A042739 Denominators of continued fraction convergents to sqrt(899). 2
1, 1, 59, 60, 3539, 3599, 212281, 215880, 12733321, 12949201, 763786979, 776736180, 45814485419, 46591221599, 2748105338161, 2794696559760, 164840505804241, 167635202364001, 9887682242916299, 10055317445280300, 593096094069173699 (list; graph; refs; listen; history; text; internal format)
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 * A172256 A172056 A032648

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

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Last modified February 16 04:47 EST 2019. Contains 320140 sequences. (Running on oeis4.)