A041479 Denominators of continued fraction convergents to sqrt(255).

1, 1, 31, 32, 991, 1023, 31681, 32704, 1012801, 1045505, 32377951, 33423456, 1035081631, 1068505087, 33090234241, 34158739328, 1057852414081, 1092011153409, 33818187016351, 34910198169760

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 = 30 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 (0,32,0,-1).

Formula

From Colin Barker, Jul 16 2012: (Start)

a(n) = 32*a(n-2) - a(n-4).

G.f.: -(x^2-x-1)/(x^4-32*x^2+1). (End)

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(30) + sqrt(34) )/2 and beta = ( sqrt(30) - sqrt(34) )/2 be the roots of the equation x^2 - sqrt(30)*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)} ( 30 + 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) = 30*a(2*n) + a(2*n - 1). (End)

Mathematica

Denominator[Convergents[Sqrt[255], 30]] (* or *) LinearRecurrence[ {0, 32, 0, -1}, {1, 1, 31, 32}, 30] (* Harvey P. Dale, Jan 19 2013 *)
CoefficientList[Series[- (x^2 - x - 1)/(x^4 - 32 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 25 2013 *)

Crossrefs

Cf. A041478, A176110, A002530.

Sequence in context: A248815 A280647 A152045 * A194380 A269267 A361263

Adjacent sequences: A041476 A041477 A041478 * A041480 A041481 A041482

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved