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A041263 Denominators of continued fraction convergents to sqrt(143). 2
1, 1, 23, 24, 551, 575, 13201, 13776, 316273, 330049, 7577351, 7907400, 181540151, 189447551, 4349386273, 4538833824, 104203730401, 108742564225, 2496540143351, 2605282707576, 59812759710023, 62418042417599, 1433009692897201, 1495427735314800 (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 = 22 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 (0,24,0,-1).

FORMULA

G.f.: (1 +x -x^2)/(x^4 -24*x^2 +1). - Vincenzo Librandi, Dec 14 2013

a(n) = 24*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 14 2013

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(22) + sqrt(26) )/2 and beta = ( sqrt(22) - sqrt(26) )/2 be the roots of the equation x^2 - sqrt(22)*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)} ( 22 + 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) = 22*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[143], 30]] (* or *) CoefficientList[Series[(1 + x - x^2)/(x^4 - 24 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 14 2013 *)

PROG

(MAGMA) I:=[1, 1, 23, 24]; [n le 4 select I[n] else 24*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 14 2013

CROSSREFS

Cf. A041262, A002530.

Sequence in context: A042094 A042096 A042098 * A042100 A042101 A042102

Adjacent sequences:  A041260 A041261 A041262 * A041264 A041265 A041266

KEYWORD

nonn,frac,easy,less

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Vincenzo Librandi, Dec 14 2013

STATUS

approved

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Last modified March 20 09:35 EDT 2019. Contains 321345 sequences. (Running on oeis4.)