|
%I #11 Dec 05 2016 10:40:04
%S 2,7,4,1,8,0,1,4,1,0,8,4,6,8,0,8,0,5,1,2,7,6,0,1,5,7,1,5,5,3,2,4,4,4,
%T 6,7,5,9,5,1,6,3,5,6,9,4,6,9,6,8,6,4,6,9,9,9,6,0,8,6,5,2,2,3,2,2,5,8,
%U 9,7,1,1,4,4,0,7,3,4,3,6,7,0,4,8,1,8,1,1,1,5,2,4,0,0,5,2,2,2,4
%N Decimal expansion of the absolute minimum of f(x)+f(2x), where f(x)=sin(x)-cos(x).
%C Let f(x)=sin(x)+cos(x) and g(x)=f(x)+f(2x)+...+f(nx), where n>=2. Then f(x) attains an absolute minimum at some x between 0 and 2*pi. Guide to related sequences (including graphs in Mathematica programs):
%C n....x.........minimum of f(x)
%C 2....A198745...A198746
%C 3....A198747...A198748
%C 4....A198749...A198750
%C 5....A198751...A198752
%C 6....A198753...A198754
%e x=5.81273216913788444549287183000...
%e min=-2.74180141084680805127601571...
%t f[t_] := Sin[t] + Cos[t]
%t x = Minimize[f[t] + f[2 t], t]
%t N[x, 30]
%t (RealDigits[N[{#1[[1]], t /. #1[[2]]}, 110]] &)[x]
%t Plot[f[t] + f[2 t], {t, -3 Pi, 3 Pi}]
%t (* Second program: *)
%t Root[27 + 162x - 207x^2 - 8x^3 + 32x^4, 1] // RealDigits[#, 10, 99]& // First (* _Jean-François Alcover_, Feb 19 2013 *)
%Y Cf. A198736.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Oct 29 2011
|
