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A346831 Table read by rows, coefficients of the characteristic polynomials of the tangent matrices. 7
1, 0, 1, -1, 0, 1, 2, -1, -2, 1, 1, 0, -6, 0, 1, 4, 9, -4, -10, 0, 1, -1, 0, 15, 0, -15, 0, 1, 14, -1, -46, 19, 34, -19, -2, 1, 1, 0, -28, 0, 70, 0, -28, 0, 1, 40, 81, -88, -196, 56, 150, -8, -36, 0, 1, -1, 0, 45, 0, -210, 0, 210, 0, -45, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

The tangent matrix M(n, k) is an N X N matrix defined with h = floor((N+1)/2) as:

   M[n - k, k + 1] = if n < h then 1 otherwise -1,

   M[N - n + k + 1, N - k] = if n < N - h then -1 otherwise 1,

   for n in [1..N-1] and for k in [0..n-1], and 0 in the main antidiagonal.

The name 'tangent matrix' derives from M(n, k) = signum(tan(Pi*(n + k)/(N + 1))) whenever the right side of this equation is defined.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

The rows with even index equal those of A135670.

The determinants of tangent matrices with even dimension are A152011.

EXAMPLE

Table starts:

[0]  1;

[1]  0,  1;

[2] -1,  0,   1;

[3]  2, -1,  -2,    1;

[4]  1,  0,  -6,    0,   1;

[5]  4,  9,  -4,  -10,   0,   1;

[6] -1,  0,  15,    0, -15,   0,   1;

[7] 14, -1, -46,   19,  34, -19,  -2,   1;

[8]  1,  0, -28,    0,  70,   0, -28,   0, 1;

[9] 40, 81, -88, -196,  56, 150,  -8, -36, 0, 1.

.

The first few tangent matrices:

1       2          3              4                  5

---------------------------------------------------------------

0;   -1  0;    1  -1  0;    1  -1  -1   0;   1   1  -1  -1   0;

      0  1;   -1   0  1;   -1  -1   0   1;   1  -1  -1   0   1;

               0   1  1;   -1   0   1   1;  -1  -1   0   1   1;

                            0   1   1  -1;  -1   0   1   1   1;

                                             0   1   1   1  -1;

MAPLE

TangentMatrix := proc(N) local M, H, n, k;

   M := Matrix(N, N); H := iquo(N + 1, 2);

   for n from 1 to N - 1 do for k from 0 to n - 1 do

       M[n - k, k + 1] := `if`(n < H, 1, -1);

       M[N - n + k + 1, N - k] := `if`(n < N - H, -1, 1);

od od; M end:

A346831Row := proc(n) if n = 0 then return 1 fi;

   LinearAlgebra:-CharacteristicPolynomial(TangentMatrix(n), x);

   seq(coeff(%, x, k), k = 0..n) end:

seq(A346831Row(n), n = 0..10);

PROG

(Julia)

using AbstractAlgebra

function TangentMatrix(N)

    M = zeros(ZZ, N, N)

    H = div(N + 1, 2)

    for n in 1:N - 1

        for k in 0:n - 1

            M[n - k, k + 1] = n < H ? 1 : -1

            M[N - n + k + 1, N - k] = n < N - H ? -1 : 1

        end

    end

M end

function A346831Row(n)

    n == 0 && return [ZZ(1)]

    R, x = PolynomialRing(ZZ, "x")

    S = MatrixSpace(ZZ, n, n)

    M = TangentMatrix(n)

    c = charpoly(R, S(M))

    collect(coefficients(c))

end

for n in 0:9 println(A346831Row(n)) end

CROSSREFS

Cf. A135670, A152011, A346837 (generalized tangent matrix).

Sequence in context: A252374 A344569 A341094 * A161780 A136571 A178562

Adjacent sequences:  A346828 A346829 A346830 * A346832 A346833 A346834

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Sep 11 2021

STATUS

approved

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Last modified May 23 04:06 EDT 2022. Contains 353959 sequences. (Running on oeis4.)