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A041479 Denominators of continued fraction convergents to sqrt(255). 2
1, 1, 31, 32, 991, 1023, 31681, 32704, 1012801, 1045505, 32377951, 33423456, 1035081631, 1068505087, 33090234241, 34158739328, 1057852414081, 1092011153409, 33818187016351, 34910198169760 (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 = 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

a(n) = 32*a(n-2)-a(n-4). G.f.: -(x^2-x-1)/(x^4-32*x^2+1). [Colin Barker, Jul 16 2012]

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 A025358

Adjacent sequences:  A041476 A041477 A041478 * A041480 A041481 A041482

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified January 20 22:57 EST 2020. Contains 331104 sequences. (Running on oeis4.)