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A041923 Denominators of continued fraction convergents to sqrt(483). 2
1, 1, 43, 44, 1891, 1935, 83161, 85096, 3657193, 3742289, 160833331, 164575620, 7073009371, 7237584991, 311051578993, 318289163984, 13679196466321, 13997485630305, 601573592939131, 615571078569436, 26455558892855443, 27071129971424879 (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 = 42 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 (0,44,0,-1).

FORMULA

G.f.: -(x^2 -x -1) / (x^4 -44*x^2 +1). - Colin Barker, Nov 27 2013

a(n) = 44*a(n-2) - a(n-4) for n>3, - Vincenzo Librandi, Dec 27 2013

From Peter Bala, May 27 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(42) + sqrt(46) )/2 and beta = ( sqrt(42) - sqrt(46) )/2 be the roots of the equation x^2 - sqrt(42)*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)} ( 42 + 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) = 42*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[483], 30]] (* Vincenzo Librandi, Dec 27 2013 *)

PROG

(MAGMA) I:=[1, 1, 43, 44]; [n le 4 select I[n] else 44*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 27 2013

CROSSREFS

Cf. A041922, A176400, A040461. A002530.

Sequence in context: A165864 A223746 A020442 * A085468 A113819 A190881

Adjacent sequences:  A041920 A041921 A041922 * A041924 A041925 A041926

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 27 2013

STATUS

approved

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Last modified November 13 12:45 EST 2019. Contains 329094 sequences. (Running on oeis4.)