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Decimal expansion of the shortest distance from x axis through (1,sqrt(2)) to y axis.

2

`%I #7 Nov 08 2022 13:09:18
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`%S 3,3,9,7,3,4,6,9,5,1,0,1,7,6,9,3,4,4,1,2,7,7,9,1,3,7,5,5,5,0,1,4,1,0,
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`%T 7,9,0,4,8,9,4,8,3,4,8,7,5,2,7,1,7,7,6,3,8,3,9,0,1,6,2,1,4,8,3,4,9,4,
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`%U 4,0,2,8,9,4,5,1,6,7,8,5,1,6,6,0,9,9,1,1,3,2,6,0,6,7,1,8,4,5,9,5
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`%N Decimal expansion of the shortest distance from x axis through (1,sqrt(2)) to y axis.
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`%C See A197008 for a discussion and guide to related sequences.
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`%e d=3.397346951017693441277913755501410790489483...
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`%e x-intercept=(2.2599...,0)
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`%e y-intercept=(0,2.5366...)
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`%p (1+2^(1/3))^(3/2) ; evalf(%) ; # _R. J. Mathar_, Nov 08 2022
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`%t f[x_] := x^2 + (k*x/(x - h))^2; t = h + (h*k^2)^(1/3);
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`%t h = 1; k = Sqrt[2]; d = N[f[t]^(1/2), 100]
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`%t RealDigits[d] (* A197031 *)
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`%t x = N[t] (* x-intercept *)
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`%t y = N[k*t/(t - h)] (* y-intercept *)
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`%t Show[Plot[k + k (x - h)/(h - t), {x, 0, t}],
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`%t ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 5}], PlotRange -> All, AspectRatio -> Automatic]
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`%Y Cf. A197008.
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`%K nonn,cons
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`%O 1,1
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`%A _Clark Kimberling_, Oct 10 2011
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