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A042511 Denominators of continued fraction convergents to sqrt(783). 2
1, 1, 55, 56, 3079, 3135, 172369, 175504, 9649585, 9825089, 540204391, 550029480, 30241796311, 30791825791, 1693000389025, 1723792214816, 94777779989089, 96501572203905, 5305862678999959, 5402364251203864, 297033532244008615, 302435896495212479 (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 = 54 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 27 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 (0,56,0,-1).

FORMULA

G.f.: -(x^2-x-1) / (x^4-56*x^2+1). - Colin Barker, Dec 16 2013

From Peter Bala, May 27 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(54) + sqrt(58) )/2 and beta = ( sqrt(54) - sqrt(58) )/2 be the roots of the equation x^2 - sqrt(54)*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)} ( 54 + 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) = 54*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[783], 30]] (* Vincenzo Librandi, Jan 23 2014 *)

CROSSREFS

Cf. A042510, A040755. A002530.

Sequence in context: A112892 A232653 A291502 * A020282 A101286 A295804

Adjacent sequences:  A042508 A042509 A042510 * A042512 A042513 A042514

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Dec 16 2013

STATUS

approved

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Last modified December 6 14:15 EST 2019. Contains 329806 sequences. (Running on oeis4.)