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Decimal expansion of least x > 0 having sin(2*x) = Pi*sin(2*Pi*x).
6

%I #13 Aug 15 2021 14:50:56

%S 4,5,9,2,9,5,3,4,1,2,6,2,1,0,7,5,5,1,0,5,4,8,3,7,5,1,0,3,5,8,0,5,2,6,

%T 4,9,1,9,2,0,0,4,0,4,2,1,4,7,2,5,5,0,7,2,7,3,4,0,9,2,9,0,5,4,5,3,3,6,

%U 9,0,1,8,8,4,9,5,0,8,0,6,9,5,5,7,2,6,2,8,7,5,9,9,6,1,8,6,1,5,9

%N Decimal expansion of least x > 0 having sin(2*x) = Pi*sin(2*Pi*x).

%C For a discussion and guide to related sequences, see A197739.

%e 0.45929534126210755105483751035805264919200...

%t b = 1; c = 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, .45, .46}, WorkingPrecision -> 110]

%t RealDigits[r] (* A197821 *)

%t m = s[r]

%t RealDigits[m] (* A197822 *)

%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, .7, .8}, WorkingPrecision -> 110]

%t RealDigits[t] (* A197823 *)

%t Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]

%t d = m/3; t = x /. FindRoot[s[x] == d, {x, .8, .9}, WorkingPrecision -> 110]

%t RealDigits[t] (* A197824 *)

%t Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]

%t d = 1; t = x /. FindRoot[s[x] == d, {x, .7, .8}, WorkingPrecision -> 110]

%t RealDigits[t] (* A197726 *)

%t Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]

%t d = 1/2; t = x /. FindRoot[s[x] == d, {x, .89, 9.1}, WorkingPrecision -> 110]

%t RealDigits[t] (* A197826 *)

%t Plot[{s[x], d}, {x, 0, Pi}, 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