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Search: id:A041011
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| A041011 |
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Denominators of continued fraction convergents to sqrt(8). |
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+0
5
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| 0, 1, 1, 5, 6, 29, 35, 169, 204, 985, 1189, 5741, 6930, 33461, 40391, 195025, 235416, 1136689, 1372105, 6625109, 7997214, 38613965, 46611179, 225058681, 271669860, 1311738121, 1583407981, 7645370045, 9228778026, 44560482149
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Sqrt(8) = 2 + continued fraction [1, 4, 1, 4, 1, 4,...] = 4/2 + 4/5 + 4/(5*29) + 4/(29*169) + 4/(169*985)... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2007
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FORMULA
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a(n) = 6*a(n-2) - a(n-4). Also a(2n) = a(2n-1)+a(2n-2), a(2n+1)=4*a(2n)+a(2n-1).
G.f.: (x+x^2-x^3)/(1-6*x^2+x^4).
a(n) = (1/4)*[3 - 2*sqrt(2)]^( - 1/4)*[3 - 2*sqrt(2)]^[(1/4)*( - 1)^n]*[3 - 2*sqrt(2)]^(1/2*n) + (1/4)*[3 + 2*sqrt(2)]^[(1/4)*( - 1)^n]*[3 + 2*sqrt(2)]^(1/2*n)*[3 + 2*sqrt(2)]^( - 1/4) - (1/4)*( - 1)^n*[3 + 2*sqrt(2)]^[(1/4)*( - 1)^n]*[3 + 2*sqrt(2)]^(1/2*n)*[3 + 2*sqrt(2)]^( - 1/4) - (1/4)*[3 - 2 *sqrt(2)]^( - 1/4)*( - 1)^n*[3 - 2*sqrt(2)]^[(1/4)*( - 1)^n]*[3 - 2*sqrt(2)]^(1/2*n) + (3/16)*[3 + 2*sqrt(2)]^[(1/4)*( - 1)^n]*[3 + 2*sqrt(2)]^(1/2*n)*[3 + 2*sqrt(2)]^( - 1/4)*sqrt(2) - (1/16)*( - 1)^n*[3 + 2*sqrt(2)]^[(1/4)*( - 1)^n]*[3 + 2*sqrt(2)]^(1/2*n)*[3 + 2*sqrt(2)]^( - 1/4) *sqrt(2) - (3/16)*[3 - 2*sqrt(2)]^( - 1/4)*sqrt(2)*[3 - 2*sqrt(2)]^[(1/4)*( - 1)^n]*[3 - 2 *sqrt(2)]^(1/2*n) + (1/16)*[3 - 2*sqrt(2)]^( - 1/4)*( - 1)^n*sqrt(2)*[3 - 2*sqrt(2)]^[(1/4)*( - 1)^n]*[3 - 2*sqrt(2)]^(1/2*n), with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Feb 19 2009]
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CROSSREFS
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Cf. A041010. Also A000129(2n)=2*A041011(2n), A000129(2n+1)=A041011(2n+1).
Sequence in context: A048060 A115761 A127040 this_sequence A152118 A041056 A042643
Adjacent sequences: A041008 A041009 A041010 this_sequence A041012 A041013 A041014
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KEYWORD
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nonn,cofr,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Entry improved by Michael Somos
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