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A042011
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Denominators of continued fraction convergents to sqrt(528).
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3
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1, 1, 45, 46, 2069, 2115, 95129, 97244, 4373865, 4471109, 201102661, 205573770, 9246348541, 9451922311, 425130930225, 434582852536, 19546776441809, 19981359294345, 898726585392989, 918707944687334, 41321876151635685, 42240584096323019, 1899907576389848521
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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 = 44 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^4-46*x^2+1). - Colin Barker, Nov 29 2013
The following remarks assume an offset of 1.
Let alpha = sqrt(11) + sqrt(12) and beta = sqrt(11) - sqrt(12) be the roots of the equation x^2 - sqrt(44)*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)} ( 44 + 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) = 44*a(2*n) + a(2*n - 1). (End)
a(2*n) = A041241(2*n) = numerator of continued fraction [4,11,4,11,...,4,11] with n pairs of 4,11. - Greg Dresden, Aug 10 2021
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MATHEMATICA
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Denominator[Convergents[Sqrt[528], 20]] (* Harvey P. Dale, Nov 14 2011 *)
CoefficientList[Series[(1 + x - x^2)/(x^4 - 46 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 12 2014 *)
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PROG
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(Magma) I:=[1, 1, 45, 46]; [n le 4 select I[n] else 46*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 12 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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