A394396 Decimal expansion of (1/2)*Sum_{k>=3} (k*tan(Pi/k)/Pi - 1).

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

Offset

0,1

Comments

This constant equals half the total fractional excess of circumscribed regular polygon half-perimeters over Pi: 2*c = Sum_{n>=3} (n*tan(Pi/n)/Pi - 1). Each term n*tan(Pi/n)/Pi - 1 measures how much the half-perimeter of the circumscribed regular n-gon exceeds Pi. The series converges since each term is asymptotic to Pi^2/(3*n^2). This constant also appears as the leading coefficient in -log(L(theta)) ~ c*(Pi/2 - theta)^2 as theta -> Pi/2, where L(0) = A379131. The inscribed analog is A394496.

Links

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

Formula

Equals (1/2)*Sum_{m>=1} T(m+1)*Pi^(2m)/((2m+1)!) * (zeta(2m) - 1 - 4^(-m)) where T(k) = A000182(k) are the tangent numbers.

Example

0.853933950843351553929946049015170139924502800848152...

Mathematica

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

Prog

(Python)
from mpmath import mp, pi, zeta, taylor
mp.dps = 80
N, tc = 2000, taylor(mp.tan, 0, 101)
c = sum((n*mp.tan(pi/n)/pi-1)/2 for n in range(3, N+1))
c += sum(tc[2*k+1]*pi**(2*k)*(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. A379131 (sinc product), A394496 (inscribed analog), A085365 (Kepler-Bouwkamp), A000182 (tangent numbers).

Sequence in context: A011466 A154509 A081885 * A019609 A334502 A093341

Adjacent sequences: A394393 A394394 A394395 * A394397 A394398 A394399

Keyword

nonn,cons

Author

Uğur Sezen, Mar 19 2026

Status

approved