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A041543 Denominators of continued fraction convergents to sqrt(288). 2
1, 1, 33, 34, 1121, 1155, 38081, 39236, 1293633, 1332869, 43945441, 45278310, 1492851361, 1538129671, 50713000833, 52251130504, 1722749176961, 1775000307465, 58522759015841, 60297759323306, 1988051057361633, 2048348816684939, 67535213191279681 (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 = 32 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 (0,34,0,-1).

FORMULA

G.f.: -(x^2-x-1) / ((x^2-6*x+1)*(x^2+6*x+1)). - Colin Barker, Nov 18 2013

a(n) = 34*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Dec 20 2013

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = 2*sqrt(2) + 3 and beta = 2*sqrt(2) - 3 be the roots of the equation x^2 - sqrt(32)*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)} ( 32 + 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) = 32*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[288], 30]] (* Vincenzo Librandi, Dec 20 2013 *)

PROG

(MAGMA) I:=[1, 1, 33, 34]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 20 2013

CROSSREFS

Cf. A041542, A040271, A002530.

Sequence in context: A115394 A260678 A140143 * A144425 A180329 A020260

Adjacent sequences:  A041540 A041541 A041542 * A041544 A041545 A041546

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 18 2013

STATUS

approved

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Last modified February 18 19:54 EST 2019. Contains 320262 sequences. (Running on oeis4.)