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A122944
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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}.
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5
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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, 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, 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
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OFFSET
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1,8
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COMMENTS
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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}}
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LINKS
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Table of n, a(n) for n=1..96.
Roger Bagula and Gary Adamson, Pascal's Triangle in Gray Code: its Hadamard and IFS
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FORMULA
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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
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EXAMPLE
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{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},
{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, 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}
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MATHEMATICA
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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]}]
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CROSSREFS
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Cf. A121801.
Sequence in context: A118210 A061399 A161856 * A107688 A060097 A098120
Adjacent sequences: A122941 A122942 A122943 * A122945 A122946 A122947
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KEYWORD
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tabl,uned,sign
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AUTHOR
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Roger L. Bagula and Gary W. Adamson, Oct 24 2006
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STATUS
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approved
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