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A123480
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Coefficients of the series giving the best rational approximations to sqrt(3).
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10
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4, 60, 840, 11704, 163020, 2270580, 31625104, 440480880, 6135107220, 85451020204, 1190179175640, 16577057438760, 230888624967004, 3215863692099300, 44791203064423200, 623860979209825504, 8689262505873133860, 121025814103014048540, 1685672134936323545704
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OFFSET
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1,1
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COMMENTS
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The partial sums of the series 2 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(3), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [1;1,2,1], [1;1,2,1,2,1], [1;1,2,1,2,1,2,1], [1;1,2,1,2,1,2,1,2,1] and so forth.
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LINKS
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FORMULA
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a(n+3) = 15*a(n+2) - 15*a(n+1) + a(n).
a(n) = -1/3 + (1/6 + 1/12*3^(1/2))*(7 + 4*3^(1/2))^n + (1/6 - 1/12*3^(1/2))*(7 - 4*3^(1/2))^n.
G.f.: -4*x/((x-1)*(x^2-14*x+1)). - Colin Barker, Jan 20 2013
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MATHEMATICA
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CoefficientList[Series[-4*x/((x - 1)*(x^2 - 14*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 13 2017 *)
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PROG
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(PARI) x='x+O('x^50); Vec(-4*x/((x-1)*(x^2-14*x+1))) \\ G. C. Greubel, Oct 13 2017 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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