A346837 -id:A346837

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A347596

Alternating row sums of A346837.

+20
1

1, -1, -2, 0, 8, 0, -32, 0, 128, 0, -512, 0, 2048, 0, -8192, 0, 32768, 0, -131072, 0, 524288, 0, -2097152, 0, 8388608, 0, -33554432, 0, 134217728, 0, -536870912, 0, 2147483648, 0, -8589934592, 0, 34359738368, 0, -137438953472, 0, 549755813888, 0, -2199023255552

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..42.

CROSSREFS

Cf. A346837.

KEYWORD

sign

AUTHOR

Peter Luschny, Sep 11 2021

STATUS

approved

A346831

Table read by rows, coefficients of the characteristic polynomials of the tangent matrices.

+10
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);

MATHEMATICA

TangentMatrix[N_] := Module[{M, H, n, k},

M = Array[0&, {N, N}]; H = Quotient[N + 1, 2];

For[n = 1, n <= N - 1, n++, For[k = 0, k <= n - 1, k++,

M[[n - k, k + 1]] = If[n < H, 1, -1];

M[[N - n + k + 1, N - k]] = If[n < N - H, -1, 1]]]; M];

A346831Row[n_] := Module[{c}, If[n == 0, Return[{1}]];

c = CharacteristicPolynomial[TangentMatrix[n], x];

(-1)^n*CoefficientList[c, x]];

Table[A346831Row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 15 2024, after Peter Luschny *)

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

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