A277445 Determinants of the equally spaced angles sines to tangents matrices.

1, -2, -4, 4, 48, -160, -32, 2176, 6912, 0, -273408, 41984, 19456, -37027840, -141705216, 0, 3833856, -34359869440, 0, 1625620480000, 11045440585728, -47710208, -520279482695680, 7719016726528, 909115392000, -207717914210467840, 0, 0, 100736516652659638272, -721057900040447590400

Offset

1,2

Links

Table of n, a(n) for n=1..30.

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.

Wikibooks, On 2D Inverse Problems/The case of the unit disc

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

Related to A007318 by the continued fraction.

Sequence in context: A360685 A290606 A155952 * A145636 A334190 A280795

Adjacent sequences: A277442 A277443 A277444 * A277446 A277447 A277448

Keyword

sign

Author

David V. Ingerman, Oct 15 2016

Status

approved