A197827 Decimal expansion of least x > 0 having sin(2*x) = 2*Pi*sin(4*Pi*x).

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, 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, 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

Offset: 0

Author

Clark Kimberling, Oct 19 2011

Keywords

• nonn
• cons

Comments

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

Examples

			0.24405505512124668685356429784849535656...
		

Crossrefs

Cf. A197739.

Programs

Mathematica

b = 1; c = 2 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, .24, .25}, WorkingPrecision -> 110]
RealDigits[r]  (* A197827 *)
m = s[r]
RealDigits[m]  (* A197828 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, .4, .42}, WorkingPrecision -> 110]
RealDigits[t] (* A197829 *)
Plot[{s[x], d}, {x, 0, .7}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, .91, .92}, WorkingPrecision -> 110]
RealDigits[t] (* A197830 *)
Plot[{s[x], d}, {x, 0, Pi/2}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[t] (* A197700 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, .93, .94}, WorkingPrecision -> 110]
RealDigits[t] (* A197832 *)
Plot[{s[x], d}, {x, 0, 1}, AxesOrigin -> {0, 0}]

Extensions

Definition corrected by Georg Fischer, Aug 10 2021