A394496 Decimal expansion of (1/2)*Sum_{n>=3} (1 - n*sin(2*Pi/n)/(2*Pi)).

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

Offset

0,3

Comments

Inscribed polygon analog of A394396. Let S_in(n) = 1 - n*sin(2*Pi/n)/(2*Pi) denote the fractional deficit of the regular n-gon inscribed in the unit circle. A394396 gives (1/2)*Sum_{n>=3} S_out(n) where S_out(n) = n*tan(Pi/n)/Pi - 1 is the circumscribed analog. For large n, S_in(n) ~ 2*Pi^2/(3*n^2), which is asymptotically twice S_out(n) ~ Pi^2/(3*n^2). The ratio S_in(n)/S_out(n) -> 2 as n -> infinity.

Links

Table of n, a(n) for n=0..99.

Formula

Equals (1/2)*Sum_{n>=3} (1 - n*sin(2*Pi/n)/(2*Pi)).

Equals Sum_{k>=1} (-1)^(k+1) * ((2*Pi)^(2*k) / (2*k+1)!) * (zeta(2*k) - 1 - 4^(-k)) / 2.

Example

1.1804856511706375818127964287912040598345365761955...

Mathematica

NSum[(1 - n Sin[2 Pi/n]/(2 Pi))/2, {n, 3, Infinity}, WorkingPrecision -> 50, NSumTerms -> 500]

Prog

(Python)
from mpmath import mp, pi, sin, zeta, fac
mp.dps = 80
N = 5000
c = sum((1 - n*sin(2*pi/n)/(2*pi))/2 for n in range(3, N+1))
c += sum((-1)**(k+1)*(2*pi)**(2*k)/fac(2*k+1)*(zeta(2*k)-sum(mp.mpf(1)/n**(2*k) for n in range(1, N+1)))/2 for k in range(1, 48))
print(c)

Crossrefs

Cf. A394396 (circumscribed analog), A394477 (log-weighted analog), A379131 (sinc product).

Sequence in context: A394602 A155528 A096152 * A393355 A021558 A335958

Adjacent sequences: A394493 A394494 A394495 * A394497 A394498 A394499

Keyword

nonn,cons

Author

Uğur Sezen, Mar 22 2026

Status

approved