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A042403
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Denominators of continued fraction convergents to sqrt(728).
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2
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1, 1, 53, 54, 2861, 2915, 154441, 157356, 8336953, 8494309, 450041021, 458535330, 24293878181, 24752413511, 1311419380753, 1336171794264, 70792352682481, 72128524476745, 3821475625473221, 3893604149949966, 206288891422871453, 210182495572821419
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OFFSET
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0,3
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COMMENTS
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The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 52 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
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LINKS
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FORMULA
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G.f.: -(x^2-x-1) / (x^4-54*x^2+1). - Colin Barker, Dec 10 2013
The following remarks assume an offset of 1.
Let alpha = sqrt(13) + sqrt(14) and beta = sqrt(13) - sqrt(14) be the roots of the equation x^2 - sqrt(52)*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)} ( 52 + 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) = 52*a(2*n) + a(2*n - 1). (End)
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MATHEMATICA
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Denominator/@Convergents[Sqrt[728], 40] (* Harvey P. Dale, May 08 2011 *)
CoefficientList[Series[(1 + x - x^2)/(x^4 - 54 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 21 2014 *)
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PROG
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(Magma) I:=[1, 1, 53, 54]; [n le 4 select I[n] else 54*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 21 2014
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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