A197821 Decimal expansion of least x > 0 having sin(2*x) = Pi*sin(2*Pi*x).

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, 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, 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

Offset: 0

Author

Clark Kimberling, Oct 19 2011

Keywords

• nonn
• cons

Comments

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

Examples

			0.45929534126210755105483751035805264919200...
		

Crossrefs

Cf. A197739.

Programs

Mathematica

b = 1; c = Pi;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .45, .46}, WorkingPrecision -> 110]
RealDigits[r]  (* A197821 *)
m = s[r]
RealDigits[m]  (* A197822 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, .7, .8}, WorkingPrecision -> 110]
RealDigits[t]   (* A197823 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, .8, .9}, WorkingPrecision -> 110]
RealDigits[t]  (* A197824 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, .7, .8}, WorkingPrecision -> 110]
RealDigits[t]  (* A197726 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, .89, 9.1}, WorkingPrecision -> 110]
RealDigits[t]  (* A197826 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]

Extensions

Definition corrected by Georg Fischer, Aug 10 2021