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A041611 Denominators of continued fraction convergents to sqrt(323). 3
1, 1, 35, 36, 1259, 1295, 45289, 46584, 1629145, 1675729, 58603931, 60279660, 2108112371, 2168392031, 75833441425, 78001833456, 2727895778929, 2805897612385, 98128414600019, 100934312212404, 3529895029821755, 3630829342034159, 126978092658983161 (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 = 34 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 28 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 (0,36,0,-1).

FORMULA

G.f.: -(x^2-x-1) / (x^4-36*x^2+1). - Colin Barker, Nov 19 2013

a(n) = 36*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Dec 21 2013

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(34) + sqrt(38) )/2 and beta = ( sqrt(34) - sqrt(38) )/2 be the roots of the equation x^2 - sqrt(34)*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)} ( 34 + 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) = 34*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[323], 30]] (* Vincenzo Librandi Dec 21 2013 *)

PROG

(MAGMA) I:=[1, 1, 35, 36]; [n le 4 select I[n] else 36*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 21 2013

CROSSREFS

Cf. A041610, A040305, A002530.

Sequence in context: A296869 A099647 A064993 * A160775 A160021 A172016

Adjacent sequences:  A041608 A041609 A041610 * A041612 A041613 A041614

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 19 2013

STATUS

approved

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Last modified November 14 12:19 EST 2019. Contains 329114 sequences. (Running on oeis4.)