A041683 Denominators of continued fraction convergents to sqrt(360).

1, 1, 37, 38, 1405, 1443, 53353, 54796, 2026009, 2080805, 76934989, 79015794, 2921503573, 3000519367, 110940200785, 113940720152, 4212806126257, 4326746846409, 159975692596981, 164302439443390, 6074863512559021, 6239165952002411

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 A345481

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