A198750 Decimal expansion of the least x>0 that gives the absolute minimum of f(x)+f(2x)+f(3x)+f(4x), where f(x)=sin(x)-cos(x).

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

Offset

1,1

Comments

See A198745 for a guide to related sequences.

Links

Table of n, a(n) for n=1..99.

Index entries for transcendental numbers.

Example

x=6.02199019639758406905296413941959795147011...
min=-5.37101677824568058580129866091440445833...

Mathematica

f[t_] := Sin[t] - Cos[t]
n = 4; s[t_] := Sum[f[k*t], {k, 1, n}]
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u] (* A198749 *)
RealDigits[v] (* A198750 *)
Plot[s[t], {t, -2 Pi, 2 Pi}, PlotRange -> {-6, 4}]
(* Alternative: *)
2*Pi + ArcSin[ Root[-5 + 5x + 160x^2 + 152x^3 - 568x^4 - 568x^5 + 512x^6 + 512x^7, 4]] // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Feb 15 2013 *)

Prog

(PARI) solve(x=6, 7, cos(x)+sin(x)+2*cos(2*x)+2*sin(2*x)+3*cos(3*x)+3*sin(3*x)+4*cos(4*x)+4*sin(4*x)) \\ Charles R Greathouse IV, May 17 2026

Crossrefs

Cf. A198745.

Sequence in context: A367309 A081823 A081802 * A322578 A087778 A129106

Adjacent sequences: A198747 A198748 A198749 * A198751 A198752 A198753

Keyword

nonn,cons

Author

Clark Kimberling, Oct 29 2011

Status

approved