OFFSET
1,2
LINKS
O. D. Biesel, D. V. Ingerman, J. A. Morrow, and W. T. Shore, Layered networks, the discrete Laplacian, and a continued fraction identity, Mathematics REU 2008, University of Washington.
Zhi-Wei Sun, On some determinants involving the tangent function, arXiv:1901.04837 [math.NT], 2019.
FORMULA
Let n be a natural number. a(n) = det(T(n)), where T(n) is the n X n matrix with entries 1,0 and -1, such that 2*T(n)*s(n) = t(n), where s(n) and t(n) are vectors of length n, given by s(n) = sin(k*Pi/(2n+1)) and t(n) = tan(k*Pi/(2n+1)), for k=1..n.
Existence of the matrix T(n) is proved for prime 2n+1, in which case the entries of T(n) are 1 and -1. Computer checked for small 2n+1...
Examples:
2*sin(Pi/3) = tan(Pi/3),
2*(-sin(Pi/5) + sin(2*Pi/5)) = tan(Pi/5),
2*(sin(Pi/5) + sin(2*Pi/5)) = tan(2*Pi/5),
2*(sin(Pi/7) + sin(2*Pi/7) - sin(3*Pi/7)) = tan(Pi/7),
2*(sin(Pi/7) + sin(2*Pi/7) + sin(3*Pi/7)) = tan(3*Pi/7),
...
EXAMPLE
a(1) = det([1]) = 1,
a(2) = det([-1 1], [1 1]) = -2,
a(3) = det([1 1 -1], [1 -1 1], [1 1 1]) = -4,
a(4) = det([-1 1 1 -1], [-1 1 -1 1], [0 0 1 0], [1 1 1 1]) = 4.
...
PROG
(SageMath)
def binary_trig(n):
N=2*n+1
print(N, "th root of unity")
T=matrix(ZZ, n, n)
for ll in range(n):
l=ll*2+1
for kk in range(n):
k=kk+1
T[min(l*n%N, N-l*n%N)-1, min(k*l%N, N-k*l%N)-1]=sign(RDF(sin(k*l*pi/N)))
s=matrix(RDF, n, 1)
for k in range(n):
s[k, 0]=sin((k+1)*pi/N)
for k in range(n):
if (T*s)[k, 0]<0:
#if prod(T[k])==0:
T[k]=-T[k]
return det(T)
CROSSREFS
KEYWORD
sign
AUTHOR
David V. Ingerman, Oct 15 2016
STATUS
approved