A336043 - OEIS

%I #16 Feb 03 2025 10:54:04

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

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

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

%N Decimal expansion of the solution s of s/c = 3/2, where s = arclength and c = chord length on the unit circle.

%C For every constant k > 1, the ratio s/c such that s/c = k is the same for all circles.  Choosing the unit circle, x^2 + y^2 = 1, we have c = sqrt(2 - 2*cos(s)), so that the solution of s/c = k is the solution of s = k*sqrt(2 - 2*cos(s)).  For each s in (0, 2*Pi) there is a unique solution c; for each c in (0, Pi), there are two solutions s; if one of them is s_0, the other is 2*Pi - s_0.

%C Guide to related sequences:

%C   ratio, s/c     arclength, s    chord length, c

%C   3/2            A336043         A336044

%C   2              A336045         A199460

%C   3              A336047         A336048

%C   Pi             A336049         A336050

%C   4              A336051         A336052

%C   5              A336053         A336054

%C   6              A336055         A336056

%C   2*Pi           A336057         A336058

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

%F s = 2.9915631364441994204309876991519194313413...

%F c = 1.9943754242961329469539917994346129542275...

%t t = t /.FindRoot[t == (3/2) Sqrt[2 - 2 Cos[t]], {t, 4}, WorkingPrecision -> 200]

%t c = 2 t/3

%t RealDigits[t][[1]]   (* A336043 *)

%t RealDigits[c][[1]]   (* A336044 *)

%o (PARI) solve(x=2,3, 1-2*x^2/9-cos(x)) \\ _Charles R Greathouse IV_, Feb 03 2025

%Y Cf. A336044.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Jul 06 2020