A197758 Decimal expansion of least x>0 having sin(2x)=4*sin(8x).

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

Offset: 0

Author

Clark Kimberling, Oct 19 2011

Keywords

• nonn
• cons

Comments

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

Examples

			x=0.37145829403348635250583272851951240980...
		

Crossrefs

Cf. A197739

Programs

Mathematica

b = 1; c = 4;
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, .37, .38}, WorkingPrecision -> 110]
RealDigits[r]  (* A197758 *)
m = s[r]
RealDigits[m]  (* A197759 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, 0.64, 0.65}, WorkingPrecision -> 110]
RealDigits[t]  (* A197760 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, 0.72, 0.73}, WorkingPrecision -> 110]
RealDigits[t]  (* A197761 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, 0.6, 0.7}, WorkingPrecision -> 110]
RealDigits[t]  (* A019692, pi/5 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, 0.6, 0.8}, WorkingPrecision -> 110]
RealDigits[t]   (* A003881 *)
Plot[{s[x], d}, {x, 0, 1}, AxesOrigin -> {0, 0}]
RealDigits[ ArcTan[ Sqrt[ Root[17#^3 - 109#^2 + 115# - 15&, 1] ] ], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)