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A123949
This sequence needs a definition.
1
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
1,5
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!
EXAMPLE
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;
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}]];
Flatten[%]
CROSSREFS
KEYWORD
uned,tabl,sign,less,uned
AUTHOR
STATUS
approved