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A041219
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Denominators of continued fraction convergents to sqrt(120).
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3
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1, 1, 21, 22, 461, 483, 10121, 10604, 222201, 232805, 4878301, 5111106, 107100421, 112211527, 2351330961, 2463542488, 51622180721, 54085723209, 1133336644901, 1187422368110, 24881784007101, 26069206375211, 546265911511321, 572335117886532, 11992968269241961
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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 = 20 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
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LINKS
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FORMULA
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a(n) = 22*a(n-2) - a(n-4).
G.f.: (1+x-x^2)/(1-22*x^2+x^4). (End)
The following remarks assume an offset of 1.
Let alpha = sqrt(5) + sqrt(6) and beta = sqrt(5) - sqrt(6) be the roots of the equation x^2 - sqrt(20)*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)} ( 20 + 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) = 20*a(2*n) + a(2*n - 1). (End)
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MATHEMATICA
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Denominator[Convergents[Sqrt[120], 30]] (* Harvey P. Dale, Mar 14 2013 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 22 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 24 2013 *)
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CROSSREFS
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KEYWORD
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nonn,frac,easy,less
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AUTHOR
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STATUS
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approved
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