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%I #14 Aug 15 2021 15:49:26
%S 2,4,4,0,5,5,0,5,5,1,2,1,2,4,6,6,8,6,8,5,3,5,6,4,2,9,7,8,4,8,4,9,5,3,
%T 5,6,5,6,6,3,6,9,3,6,1,6,5,8,4,1,3,6,0,5,9,4,5,7,7,6,9,0,2,8,3,2,8,3,
%U 5,3,4,7,3,8,2,2,4,7,1,9,2,5,0,9,7,7,9,7,3,9,6,8,9,3,1,4,0,6,6
%N Decimal expansion of least x > 0 having sin(2*x) = 2*Pi*sin(4*Pi*x).
%C For a discussion and guide to related sequences, see A197739.
%e 0.24405505512124668685356429784849535656...
%t b = 1; c = 2 Pi;
%t f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
%t r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .24, .25}, WorkingPrecision -> 110]
%t RealDigits[r] (* A197827 *)
%t m = s[r]
%t RealDigits[m] (* A197828 *)
%t Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
%t d = m/2; t = x /. FindRoot[s[x] == d, {x, .4, .42}, WorkingPrecision -> 110]
%t RealDigits[t] (* A197829 *)
%t Plot[{s[x], d}, {x, 0, .7}, AxesOrigin -> {0, 0}]
%t d = m/3; t = x /. FindRoot[s[x] == d, {x, .91, .92}, WorkingPrecision -> 110]
%t RealDigits[t] (* A197830 *)
%t Plot[{s[x], d}, {x, 0, Pi/2}, AxesOrigin -> {0, 0}]
%t d = 1; t = x /. FindRoot[s[x] == d, {x, .4, .5}, WorkingPrecision -> 110]
%t RealDigits[t] (* A197700 *)
%t Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t d = 1/2; t = x /. FindRoot[s[x] == d, {x, .93, .94}, WorkingPrecision -> 110]
%t RealDigits[t] (* A197832 *)
%t Plot[{s[x], d}, {x, 0, 1}, AxesOrigin -> {0, 0}]
%Y Cf. A197739.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Oct 19 2011
%E Definition corrected by _Georg Fischer_, Aug 10 2021
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