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A041683 Denominators of continued fraction convergents to sqrt(360). 2
1, 1, 37, 38, 1405, 1443, 53353, 54796, 2026009, 2080805, 76934989, 79015794, 2921503573, 3000519367, 110940200785, 113940720152, 4212806126257, 4326746846409, 159975692596981, 164302439443390, 6074863512559021, 6239165952002411 (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 = 36 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 (0,38,0,-1).

FORMULA

G.f.: -(x^2-x-1) / ((x^2-6*x-1)*(x^2+6*x-1)). - Colin Barker, Nov 21 2013

a(n) = 38*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Dec 22 2013

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = 3 + sqrt(10) and beta = 3 - sqrt(10) be the roots of the equation x^2 - 6*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) = A005668(n+1) for n even; a(n) = 1/6*A005668(n+1) for n odd.

a(n) = product {k = 1..floor((n-1)/2)} ( 36 + 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) = 36*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[360], 30]] (* Vincenzo Librandi, Dec 22 2013 *)

PROG

(MAGMA) I:=[1, 1, 37, 38]; [n le 4 select I[n] else 38*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 22 2013

CROSSREFS

Cf. A041682, A040341, A002530, A005668.

Sequence in context: A071887 A168143 A111043 * A064172 A242989 A295801

Adjacent sequences:  A041680 A041681 A041682 * A041684 A041685 A041686

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 21 2013

STATUS

approved

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)