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Search: id:A122947
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| A122947 |
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Binary Hadamard Matrix self-Similarity of the Pscal triangle type used to give characteristic polynomials as a triangular sequence: 1, 1 - x, -1 - x + x^2, 1 + 3 x + x^2 - x^3, 1 - x - 4 x^2 - x^3 +x^4, -1 - x + 6 x^2 + 8 x^3 + x^4 - x^5, -1 + 3 x + 6 x^2 - 7 x^3 - 10 x^4 - x^5 + x^6. |
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| 1, 1, -1, -1, -1, 1, 1, 3, 1, -1, 1, -1, -4, -1, 1, -1, -1, 6, 8, 1, -1, -1, 3, 6, -7, -10, -1, 1, -1, 7, -6, -14, 8, 12, 1, -1, 1, 1, -13, 8, 20, -8, -13, -1, 1, -1, -3, 11, 17, -36, -26, 27, 21, 1, -1, -1, 1, 19, -26, -53, 63, 53, -32, -25, -1, 1, -1, 5, 15, -89, 54, 144, -102, -92, 37, 29, 1, -1, 1, 3, -27, -10, 169, -89, -226, 127, 121
(list; table; graph; listen)
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OFFSET
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1,8
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COMMENT
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Matrices: 1 X 1 {{1}} 2 X 2 {{1, 1}, {1, 0}} 3 X 3 {{1, 1, 1}, {1, 0, 1}, {1, 1, 0}} 4 X 4 {{1, 1, 1, 1}, {1, 0, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}} 5 X 5 {{1, 1, 1, 1, 1}, {1, 0, 1, 0, 1}, {1, 1, 0, 0, 1}, {1, 0, 0, 0, 1}, {1, 1, 1, 1, 0}} 6 X 6 {{1, 1, 1, 1, 1, 1}, {1, 0, 1, 0, 1, 0}, {1, 1, 0, 0, 1, 1}, {1, 0, 0, 0, 1, 0}, {1, 1, 1, 1, 0, 0}, {1, 0, 1, 0, 0, 0}}
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FORMULA
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c(i,k)=Floor[Mod[i/2^k,2]] a(i,j)=If[Sum[c(i,k)*c(j,k),{k,0,d-1}]==0,1,0] a(i,j)->p(n,x) p(n,x)->{t(n,m)
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EXAMPLE
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Triangle sequence:
{1},
{1, -1},
{-1, -1, 1},
{1, 3, 1, -1},
{1, -1, -4, -1, 1},
{-1, -1, 6, 8, 1, -1},
{-1, 3, 6, -7, -10, -1, 1},
{-1, 7, -6, -14, 8, 12, 1, -1}
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MATHEMATICA
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Clear[b, c, An] c[i_, k_] := Floor[Mod[i/2^k, 2]]; An[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}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]
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CROSSREFS
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Sequence in context: A052371 A062278 A016465 this_sequence A046534 A131324 A079724
Adjacent sequences: A122944 A122945 A122946 this_sequence A122948 A122949 A122950
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KEYWORD
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tabl,uned,sign
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AUTHOR
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Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 24 2006
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