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A041263
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Denominators of continued fraction convergents to sqrt(143).
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2
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1, 1, 23, 24, 551, 575, 13201, 13776, 316273, 330049, 7577351, 7907400, 181540151, 189447551, 4349386273, 4538833824, 104203730401, 108742564225, 2496540143351, 2605282707576, 59812759710023, 62418042417599, 1433009692897201, 1495427735314800
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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 = 22 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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The following remarks assume an offset of 1.
Let alpha = ( sqrt(22) + sqrt(26) )/2 and beta = ( sqrt(22) - sqrt(26) )/2 be the roots of the equation x^2 - sqrt(22)*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)} ( 22 + 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) = 22*a(2*n) + a(2*n - 1). (End)
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MATHEMATICA
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Denominator[Convergents[Sqrt[143], 30]] (* or *) CoefficientList[Series[(1 + x - x^2)/(x^4 - 24 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 14 2013 *)
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PROG
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(Magma) I:=[1, 1, 23, 24]; [n le 4 select I[n] else 24*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 14 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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EXTENSIONS
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STATUS
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approved
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