A197589 Decimal expansion of least x>0 satisfying f(x)=m/2, where m is the maximal value of the function f(x)=cos(x)^2+sin(2x)^2.

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

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..98.

Example

x=1.12868019433775284470060498453346294726...

Mathematica

b = 1; c = 2;
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, .65, .66}, WorkingPrecision -> 110]
RealDigits[r]  (* A195700, arcsin(sqrt(3/8)) *)
m = s[r]
RealDigits[m]
Rationalize[{m, m/2, m/3, 2 m/3, m/4, 3 m/4}]
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t]  (* A197589 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t]  (* A197591 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t] (* A019670, pi/3 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t]  (* A197592 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]

Crossrefs

Cf. A197739, A195700.

Sequence in context: A019914 A065473 A054029 * A124356 A277249 A320262

Adjacent sequences: A197586 A197587 A197588 * A197590 A197591 A197592

Keyword

nonn,cons

Author

Clark Kimberling, Oct 18 2011

Status

approved