A197700 - OEIS

%I #13 Mar 31 2023 17:28:14

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

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

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

%N Decimal expansion of Pi/(1 + 2*Pi).

%C Least x > 0 such that sin(b*x) = cos(c*x) (and also sin(c*x) = cos(b*x)), where b=1/2 and c=Pi; see the Mathematica program for a graph and A197682 for a discussion and guide to related sequences.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 0.4313487191507935144267938371456753323979...

%t b = 1/2; c = Pi;

%t t = x /. FindRoot[Sin[b*x] == Cos[c*x], {x, .43, .44}]

%t N[Pi/(2*b + 2*c), 110]

%t RealDigits[%]  (* A197700 *)

%t Simplify[Pi/(2*b + 2*c)]

%t Plot[{Sin[b*x], Cos[c*x]}, {x, 0, 1}]

%t RealDigits[Pi/(1+2 Pi),10,120][[1]] (* _Harvey P. Dale_, Mar 31 2023 *)

%o (PARI) 1/(1/Pi+2) \\ _Charles R Greathouse IV_, Sep 30 2022

%Y Cf. A197682.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 17 2011