A123949 This sequence needs a definition.

1, 1, -1, 1, -2, 1, 0, -2, 3, -1, 1, 0, -2, 0, 1, 0, -2, -1, 3, 1, -1, 0, 0, -3, 6, -2, -2, 1, 0, 2, -9, 15, -11, 3, 1, -1, 1, -4, 2, 6, -1, -6, -1, 2, 1, 0, -2, 7, -1, -11, -3, 8, 4, -1, -1, 0, 0, -3, -6, 4, 18, -9, -2, -3, 0, 1, 0, 0, 0, -4, 3, 19, -29, 11, -2, 2, 1, -1, 0, 0, 0, 0, 4, 0, -25, 16, 26, -20, -4, 2, 1, 0, 0, 0, -4, 11, 7, -63, 63, 8, -34

Offset: 1

Author

Gary W. Adamson and Roger L. Bagula, Oct 26 2006

Keywords

• uned
• tabl
• sign
• less

Comments

n-th level Hadamard matrices for Pascal type binary are inverted and multiplied by the corresponding n th level Gray code Hadamard (some of which matrices have determinant zero): the resulting matrix is processed to give a triangular sequence.

Matrices:

1 X 1 {{1}}

2 X 2 {{1, 0}, {0, 1}}

3 X 3 {{1, -1, -1}, {0, 1, 1}, {0, 1, 1}}

4 X 4 {{1, 1, 0, 0}, {0, -1, 0, 0}, {0, -1, 0, 1}, {0, 2, 1, 0}}

5 X 5 {{1, 1, -1, -1, -1}, {0, -1, 0, 0, 0}, {0, -1, 0, 1, 1}, {0, 2, 1, 0, 0}, {0, 0, 1, 1, 1}}

6 X 6 {{1, 1, 0, -1, -1, 0}, {0, -1, -1, 0, 0, -1}, {0, -1, 0, 1, 1, 0}, {0, 2, 1, 0, 0, 1}, {0, 0, 0, 1, 1, 0}. {0, 0, 1, 0, 0, 1}}.

They don't get interesting until 4 X 4!

Examples

			Triangle begins
  1;
  1, -1;
  1, -2, 1;
  0, -2, 3, -1;
  1, 0, -2, 0, 1;
  0, -2, -1, 3,1, -1;
  0, 0, -3, 6, -2, -2, 1;
  0, 2, -9, 15, -11,3, 1, -1;
  1, -4, 2, 6, -1, -6, -1, 2, 1;
  0, -2,7, -1, -11, -3, 8, 4, -1, -1;
  0, 0, -3, -6, 4, 18, -9, -2, -3, 0, 1;
Polynomials:
  1;
  1 -  x;
  1 - 2x +  x^2;
  0 - 2x + 3x^2 -   x^3;
  1 + 0x - 2x^2         +   x^4;
  0 - 2x -  x^2 +  3x^3 +   x^4 -  x^5;
  0 + 0x + 3x^2 +  6x^3 -  2x^4 - 2x^5 + x^6;
  0 + 2x - 9x^2 + 15x^3 - 11x^4 + 3x^5 + x^6 -  x^7;
  1 - 4x + 2x^2 +  6x^3 -   x^4 - 6x^5 - x^6 + 2x^7 + x^8;
		

Crossrefs

Cf. A122944, A121801, A122947.

Programs

Mathematica

c[i_, k_] := Floor[Mod[i/2^k, 2]];
b[i_, k_] := If[c[i, k] == 0 && c[ i, k + 1] == 0, 0, If[c[i, k] == 1 && c[i, k + 1] == 1, 0, 1]];
An[d_] := Table[If[Sum[b[n, k]*b[m, k], {k, 0, d - 1}] == 0, 1, 0], {n, 0, d - 1}, {m, 0, d - 1}];
Bn[d_] := Table[If[Sum[c[n, k]*c[ m, k], {k, 0, d - 1}] == 0, 1, 0], {n, 0, d - 1}, {m, 0, d - 1}];
Xn[d_] := MatrixPower[Bn[d], -1].An[d];
a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[Xn[d], x], x], {d, 1, 20}]];
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