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A122944 Hadamard self-similarity matrices modulo two that are closely related to Pascal's triangle are translated to Gray code matrices: the result has characteristic polynomials that give a triangular sequence: the absolute value of the row sum is:{1, 2, 3, 4, 8, 12, 15, 26, 66, 106, 147, 182, 252, 558, 864, 1972, 5912, 9852, 14656, 19410, 28748}. 5

%I

%S 1,1,-1,-1,-1,1,0,2,1,-1,1,-1,-4,-1,1,0,-2,2,6,1,-1,0,0,4,-2,-7,-1,1,

%T 0,2,-1,-9,3,9,1,-1,1,1,-13,8,20,-8,-13,-1,1,0,-2,-2,24,-15,-31,13,17,

%U 1,-1,0,0,4,4,-40,20,44,-14,-19,-1,1,0,0,0,-8,-4,56,-24,-54,14,20,1,-1,0,0,0,0,16,8,-88,30,71,-15,-22,-1,1,0,0,0,16,8

%N Hadamard self-similarity matrices modulo two that are closely related to Pascal's triangle are translated to Gray code matrices: the result has characteristic polynomials that give a triangular sequence: the absolute value of the row sum is:{1, 2, 3, 4, 8, 12, 15, 26, 66, 106, 147, 182, 252, 558, 864, 1972, 5912, 9852, 14656, 19410, 28748}.

%C 1 X 1 {{1}}, 2 X 2 {{1, 1}, {1, 0}}, 3 X 3 {{1,1, 1}, {1, 0, 0}, {1, 0, 0}}, 4 X 4 {{1, 1, 1, 1}, {1, 0, 0, 1}, {1, 0,0, 0}, {1, 1, 0, 0}}, 5 X 5 {{1, 1, 1, 1, 1}, {1, 0, 0, 1, 1}, {1, 0, 0, 0, 0}, {1, 1, 0, 0, 0}, {1, 1, 0, 0, 0}}, 6 X 6 {{1, 1, 1, 1, 1, 1}, {1, 0, 0, 1, 1, 0}, {1, 0, 0, 0, 0, 0}, {1,1, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0}}

%H Roger Bagula and Gary Adamson, <a href="http://paulbourke.net/fractals/pascaltriangle/">Pascal's Triangle in Gray Code: its Hadamard and IFS</a>

%F Binary Matrix: b(i,j) matrix multiplication of two: a[i,j]=b[i,k).b[j,k] a[i,j]-> p[n,x] p(n,x)->t(n,m] Polynomials: 1, 1 - x, -1 - x + x^2, 2 x + x^2 - x^3, 1 - x - 4 x^2 - x^3 + x^4, -2 x + 2 x^2 + 6 x^3 +x^4 - x^5, 4 x^2 - 2 x^3 - 7 x^4 - x^5 +x^6, 2 x - x^2 - 9 x^3 + 3 x^4 + 9 x^5 + x^6 - x^7

%e {1},

%e {1, -1},

%e {-1, -1, 1},

%e {0, 2, 1, -1},

%e {1, -1, -4, -1, 1},

%e {0, -2, 2,6, 1, -1},

%e {0, 0, 4, -2, -7, -1, 1},

%e {0, 2, -1, -9, 3, 9,1, -1},

%e {1, 1, -13, 8, 20, -8, -13, -1, 1},

%e {0, -2, -2, 24, -15, -31, 13,17, 1, -1},

%e {0, 0, 4, 4, -40, 20, 44, -14, -19, -1, 1},

%e {0, 0, 0, -8, -4, 56, -24, -54, 14, 20, 1, -1}

%t 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}]: a=Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[a] RowSum=Table[Apply[Plus, Abs[a[[n]]]], {n, 1, Length[a]}]

%Y Cf. A121801.

%K tabl,uned,sign

%O 1,8

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 24 2006

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Last modified October 18 17:13 EDT 2019. Contains 328186 sequences. (Running on oeis4.)