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A042103 Denominators of continued fraction convergents to sqrt(575). 2
1, 1, 47, 48, 2255, 2303, 108193, 110496, 5191009, 5301505, 249060239, 254361744, 11949700463, 12204062207, 573336561985, 585540624192, 27508205274817, 28093745899009, 1319820516629231, 1347914262528240, 63323876592928271, 64671790855456511 (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 = 46 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 (0,48,0,-1).

FORMULA

G.f.: -(x^2-x-1) / (x^4-48*x^2+1). - Colin Barker, Dec 01 2013

a(n) = 48*a(n-2) - a(n-4) for n > 3. - Vincenzo Librandi, Jan 14 2014

From Peter Bala, May 27 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(46) + sqrt(50) )/2 and beta = ( sqrt(46) - sqrt(50) )/2 be the roots of the equation x^2 - sqrt(46)*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)} ( 46 + 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) = 46*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[575], 30]] (* Vincenzo Librandi, Jan 14 2014 *)

PROG

(MAGMA) I:=[1, 1, 47, 48]; [n le 4 select I[n] else 48*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 14 2014

CROSSREFS

Cf. A042102, A040551, A002530.

Sequence in context: A165868 A291513 A249372 * A084366 A257785 A234023

Adjacent sequences:  A042100 A042101 A042102 * A042104 A042105 A042106

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Colin Barker, Dec 01 2013

STATUS

approved

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Last modified April 18 10:56 EDT 2021. Contains 343087 sequences. (Running on oeis4.)