Revision History for A197140
(Underlined text is an addition;
strikethrough text is a deletion.)
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#9 by R. J. Mathar at Tue Nov 08 12:14:56 EST 2022
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#8 by R. J. Mathar at Tue Nov 08 12:14:51 EST 2022
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| COMMENTS
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A root of the polynomial 2*x^3-4*x^2+3*x-2. - R. J. Mathar, Nov 08 2022
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#7 by R. J. Mathar at Tue Nov 08 11:31:36 EST 2022
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| DATA
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1, 4, 4, 0, 6, 1, 9, 7, 0, 0, 5, 3, 8, 1, 9, 9, 1, 1, 7, 6, 3, 3, 2, 5, 2, 3, 0, 2, 5, 8, 9, 2, 7, 7, 4, 3, 5, 3, 7, 9, 9, 0, 9, 4, 7, 2, 6, 0, 8, 9, 0, 3, 3, 7, 7, 3, 9, 8, 4, 6, 7, 3, 6, 4, 2, 5, 6, 5, 6, 3, 7, 3, 8, 9, 3, 2, 7, 7, 8, 9, 2, 9, 4, 2, 8, 1, 7, 1, 4, 8, 8, 0, 4, 1, 0, 3, 9, 7, 9, 2
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| EXTENSIONS
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Incorrect trailing digits removed. - R. J. Mathar, Nov 08 2022
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| STATUS
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approved
editing
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#6 by Russ Cox at Fri Mar 30 18:57:52 EDT 2012
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| AUTHOR
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_Clark Kimberling (ck6(AT)evansville.edu), _, Oct 11 2011
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Discussion
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| Fri Mar 30
| 18:57
| OEIS Server: https://oeis.org/edit/global/285
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#5 by T. D. Noe at Tue Oct 11 15:27:16 EDT 2011
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#4 by Clark Kimberling at Tue Oct 11 15:21:13 EDT 2011
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#3 by Clark Kimberling at Tue Oct 11 12:32:37 EDT 2011
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| MATHEMATICA
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N[m*k*t/(k + m*t - m*h)]} (* endpt. on line y=2x *)
ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 1.7}, AspectRatio -> Automatic]
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| CROSSREFS
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Cf. A197032, A197041A197141, A197008, A195284.
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#2 by Clark Kimberling at Tue Oct 11 11:42:38 EDT 2011
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| NAME
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allocatedDecimal expansion of the x-intercept of the shortest segment from the x axis through (1,1) to forthe Clarkline Kimberlingy=2x.
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| DATA
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1, 4, 4, 0, 6, 1, 9, 7, 0, 0, 5, 3, 8, 1, 9, 9, 1, 1, 7, 6, 3, 3, 2, 5, 2, 3, 0, 2, 5, 8, 9, 2, 7, 7, 4, 3, 5, 3, 7, 9, 9, 0, 9, 4, 7, 2, 6, 0, 8, 9, 0, 3, 3, 7, 7, 3, 9, 8, 4, 6, 7, 3, 6, 4, 2, 5, 6, 5, 6, 3, 7, 3, 8, 9, 3, 2, 7, 7, 8, 9, 2, 9, 4, 2, 8, 1, 7, 1, 4, 8, 8, 0, 4, 1, 0, 3, 9, 7, 9, 2
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| OFFSET
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1,2
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| COMMENTS
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The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T. For discussions and guides to related sequences, see A197032, A197008 and A195284.
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| EXAMPLE
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length of Philo line: 1.6736473041529...; see A197139
endpoint on x axis: (1.44062, 0)
endpoint on line y=2x: (0.765782, 1.53156)
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| MATHEMATICA
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f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
g[t_] := D[f[t], t]; Factor[g[t]]
p[t_] := h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 - 3 h m t^2 + m t^3 m = 2; h = 1; k = 1; (* slope m, point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t] (* A197140 *)
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t - m*h)],
N[m*k*t/(k + m*t - m*h)]} (* endpt. on line y=2x *)
d = N[Sqrt[f[t]], 100]
RealDigits[d] (* A197141 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],
ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 1.7}, AspectRatio -> Automatic]
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| CROSSREFS
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Cf. A197032, A197041, A197008, A195284.
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| KEYWORD
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allocated
nonn,cons
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| AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu), Oct 11 2011
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| STATUS
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approved
editing
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#1 by Clark Kimberling at Mon Oct 10 19:42:39 EDT 2011
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| NAME
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allocated for Clark Kimberling
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| KEYWORD
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allocated
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| STATUS
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approved
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