

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
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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 = 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

From Colin Barker, Jul 16 2012: (Start)
a(n) = 32*a(n2)  a(n4).
G.f.: (x^2x1)/(x^432*x^2+1). (End)
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((n1)/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



