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A041363 Denominators of continued fraction convergents to sqrt(195). 2
1, 1, 27, 28, 755, 783, 21113, 21896, 590409, 612305, 16510339, 17122644, 461699083, 478821727, 12911063985, 13389885712, 361048092497, 374437978209, 10096435525931, 10470873504140, 282339146633571, 292810020137711, 7895399670214057, 8188209690351768 (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 = 26 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 (0,28,0,-1).

FORMULA

G.f.: -(x^2-x-1) / (x^4-28*x^2+1). - Colin Barker, Nov 16 2013

a(n) = 28*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 16 2013

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(26) + sqrt(30) )/2 and beta = ( sqrt(26) - sqrt(30) )/2 be the roots of the equation x^2 - sqrt(26)*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)} ( 26 + 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) = 26*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[195], 30]] (* Vincenzo Librandi, Dec 16 2013 *)

PROG

(MAGMA) I:=[1, 1, 27, 28]; [n le 4 select I[n] else 28*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 16 2013

CROSSREFS

Cf. A041362, A040181, A002530.

Sequence in context: A042502 A042504 A042506 * A042508 A042509 A042510

Adjacent sequences:  A041360 A041361 A041362 * A041364 A041365 A041366

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 16 2013

STATUS

approved

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Last modified December 10 04:15 EST 2019. Contains 329885 sequences. (Running on oeis4.)