A336043 Decimal expansion of the solution s of s/c = 3/2, where s = arclength and c = chord length on the unit circle.

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

Offset

1,1

Comments

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.

Guide to related sequences:

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

3/2 A336043 A336044

2 A336045 A199460

3 A336047 A336048

Pi A336049 A336050

4 A336051 A336052

5 A336053 A336054

6 A336055 A336056

2*Pi A336057 A336058

Links

Table of n, a(n) for n=1..86.

Formula

s = 2.9915631364441994204309876991519194313413...

c = 1.9943754242961329469539917994346129542275...

Mathematica

t = t /.FindRoot[t == (3/2) Sqrt[2 - 2 Cos[t]], {t, 4}, WorkingPrecision -> 200]
c = 2 t/3
RealDigits[t][[1]]   (* A336043 *)
RealDigits[c][[1]]   (* A336044 *)

Crossrefs

Cf. A336044.

Sequence in context: A011072 A175295 A198141 * A340723 A201765 A160331

Adjacent sequences: A336040 A336041 A336042 * A336044 A336045 A336046

Keyword

nonn,cons

Author

Clark Kimberling, Jul 06 2020

Status

approved