A041219 Denominators of continued fraction convergents to sqrt(120).

1, 1, 21, 22, 461, 483, 10121, 10604, 222201, 232805, 4878301, 5111106, 107100421, 112211527, 2351330961, 2463542488, 51622180721, 54085723209, 1133336644901, 1187422368110, 24881784007101, 26069206375211, 546265911511321, 572335117886532, 11992968269241961

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 = 20 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 (0,22,0,-1).

Formula

From Colin Barker, Jul 15 2012: (Start)

a(n) = 22*a(n-2) - a(n-4).

G.f.: (1+x-x^2)/(1-22*x^2+x^4). (End)

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = sqrt(5) + sqrt(6) and beta = sqrt(5) - sqrt(6) be the roots of the equation x^2 - sqrt(20)*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)} ( 20 + 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) = 20*a(2*n) + a(2*n - 1). (End)

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[120], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[120], 30]] (* Harvey P. Dale, Mar 14 2013 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 22 x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 24 2013 *)

Crossrefs

Cf. A041218, A002530.

Sequence in context: A041916 A041918 A068325 * A041920 A041921 A041922

Adjacent sequences: A041216 A041217 A041218 * A041220 A041221 A041222

Keyword

nonn,frac,easy,less

Author

N. J. A. Sloane

Status

approved