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A041543
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Denominators of continued fraction convergents to sqrt(288).
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2
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1, 1, 33, 34, 1121, 1155, 38081, 39236, 1293633, 1332869, 43945441, 45278310, 1492851361, 1538129671, 50713000833, 52251130504, 1722749176961, 1775000307465, 58522759015841, 60297759323306, 1988051057361633, 2048348816684939, 67535213191279681
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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 = 32 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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G.f.: -(x^2-x-1) / ((x^2-6*x+1)*(x^2+6*x+1)). - Colin Barker, Nov 18 2013
The following remarks assume an offset of 1.
Let alpha = 2*sqrt(2) + 3 and beta = 2*sqrt(2) - 3 be the roots of the equation x^2 - sqrt(32)*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)} ( 32 + 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) = 32*a(2*n) + a(2*n - 1). (End)
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MATHEMATICA
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PROG
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(Magma) I:=[1, 1, 33, 34]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 20 2013
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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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