A144475 A triangle sequence of determinants: a(n)=If[Mod[n, 3] == 0, 1, If[Mod[n, 3] == 1, -1, If[Mod[n, 3] == 2, 0]]]; b(n,m)=If[m < n && Mod[m, 3] == 0, 0, If[m < n && Mod[m, 3] == 1, 0, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 0, 1, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 1, -1, If[m == n, -1, 0]]]]]; M={{a(m), b(n, m)}, {a(n), b(n, n)}}; t(n,m)=Det[M].

-1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, 0, -1, 1, -1, 1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1, -1, -1, 1, 0, -1, 1, 0, -1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1

Offset

1,1

Comments

Row sums are:{-1, 0, 1, 0, 0, -2, -3, 0, 3, 2}.

It took me a while to get the projection right.

The example three matrices are:

Table[M /. n -> 4, {m, 1, 3}]

M1={{-1, 0},

{-1, -1}};

M2={{0, 1},

{-1, -1}};

M3={{1, 0},

{-1, -1}};

Characteristic polynomials:

Table[CharacteristicPolynomial[M /. n -> 4, x], {m, 1, 3}];

{1 + 2 x + x^2, 1 + x + x^2, -1 + x^2}.

Links

Table of n, a(n) for n=1..55.

Formula

a(n)=If[Mod[n, 3] == 0, 1, If[Mod[n, 3] == 1, -1, If[Mod[n, 3] == 2, 0]]]; b(n,m)=If[m < n && Mod[m, 3] == 0, 0, If[m < n && Mod[m, 3] == 1, 0, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 0, 1, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 1, -1, If[m == n, -1, 0]]]]]; M={{a(m), b(n, m)}, {a(n), b(n, n)}}; t(n,m)=Det[M].

Example

{-1},
{-1, 1},
{-1, 1, 1},
{-1, 1, 1, -1},
{-1, 1, 0, -1, 1},
{-1, 1, -1, -1, 1, -1},
{-1, 1, -1, -1, 1, -1, -1},
{-1, 1, 0, -1, 1, 0, -1, 1},
{-1, 1, 1, -1, 1, 1, -1, 1, 1},
{-1, 1, 1, -1, 1, 1, -1, 1, 1, -1}

Mathematica

Clear[a, b, t, n, m] a[n_] := If[Mod[n, 3] == 0, 1, If[Mod[n, 3] == 1, -1, If[Mod[n, 3] == 2, 0]]]; b[n_, m_] := If[m < n && Mod[m, 3] == 0, 0, If[m < n && Mod[m, 3] == 1, 0, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 0, 1, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 1, -1, If[m == n, -1, 0]]]]]; M := {{a[m], b[n, m]}, {a[n], b[n, n]}}; t[n_, m_] := Det[M]; Table[Table[t[n, m], {m, 0, n - 1}], {n, 1, 10}]; Flatten[%]

Crossrefs

Sequence in context: A108358 A267959 A144384 * A011758 A015088 A015166

Adjacent sequences: A144472 A144473 A144474 * A144476 A144477 A144478

Keyword

sign,uned

Author

Roger L. Bagula and Gary W. Adamson, Oct 10 2008

Status

approved