

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
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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^2x1) / (x^456*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((n1)/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



