OFFSET
0,1
COMMENTS
This constant is a special case of a general identity. For any odd integer k >= 3, define the infinite product P(k) = Product_{m>=0} f((2*m+1)*k), where:
f(n) = (n-1)/(n+1) if n == 1 (mod 4);
f(n) = (n+1)/(n-1) if n == 3 (mod 4).
This product converges to tan(j*Pi/k), where j = floor((k+1)/4).
The sequence of numerators j for k = 3, 5, 7, 9, 11, 13, ... is 1, 1, 2, 2, 3, 3, ... (A004526).
Equivalently, the product can be written using the Dirichlet character modulo 4 (A010892) as:
P(k) = Product_{m>=0} (( (2*m+1)*k - 1 ) / ( (2*m+1)*k + 1 ))^(-chi4((2*m+1)*k));
For k = 13, j = floor(14/4) = 3, so the product converges to tan(3*Pi/13).
Also, a root of the polynomial x^12 - 78*x^10 + 715*x^8 - 1716*x^6 + 1287*x^4 - 286*x^2 + 13.
LINKS
FORMULA
Equals tan(3*Pi/13).
Equals Sum_{n>=1} -(((-4)^n * (4^n - 1) * B_{2*n} * (3/13)^(2*n-1)) / (2*n)! ) * Pi^(2*n-1).
Equals Product_{m>=0} f((2*m+1)*13), where f(n) = (n-1)/(n+1) if n == 1 (mod 4) and f(n) = (n+1)/(n-1) if n == 3 (mod 4).
Equals (2 * i^(25/13) + i^(11/13) - i) / (i^(24/13) - 1) where i is the imaginary unit. - Michal Paulovic, Apr 28 2026
EXAMPLE
0.885922693643422280142121178600396087556...
MATHEMATICA
RealDigits[Tan[3*Pi/13], 10, 120][[1]] (* Amiram Eldar, Apr 18 2026 *)
PROG
(PARI) tan(3*Pi/13)
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Dimitris Valianatos, Apr 18 2026
STATUS
approved