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A041839
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Denominators of continued fraction convergents to sqrt(440).
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2
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1, 1, 41, 42, 1721, 1763, 72241, 74004, 3032401, 3106405, 127288601, 130395006, 5343088841, 5473483847, 224282442721, 229755926568, 9414519505441, 9644275432009, 395185536785801, 404829812217810, 16588378025498201, 16993207837716011
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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 = 40 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^4-42*x^2+1). - Colin Barker, Nov 25 2013
The following remarks assume an offset of 1.
Let alpha = sqrt(10) + sqrt(11) and beta = sqrt(10) - sqrt(11) be the roots of the equation x^2 - sqrt(40)*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)} ( 40 + 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) = 40*a(2*n) + a(2*n - 1). (End)
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MATHEMATICA
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Denominator[Convergents[Sqrt[440], 20]] (* Harvey P. Dale, Feb 21 2013 *)
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PROG
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(Magma) I:=[1, 1, 41, 42]; [n le 4 select I[n] else 42*Self(n-2)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Dec 25 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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