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Search: id:A143209
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| A143209 |
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A "completed" set of cyclotomic polynomial with coefficients that are a triangular sequence: ( filled out with powers of (x+1)^m) p(x,n)=If[PrimeQ[n], Cyclotomic[n, x]*(x + 1), Cyclotomic[n, x]*(x + 1)^(n + 1 - Length[CoefficientList[Cyclotomic[n, x], x]])];. |
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+0
1
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| 1, -1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 3, 2, 3, 3, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 4, 6, 4, 2, 4, 6, 4, 1, 1, 3, 3, 2, 3, 3, 2, 3, 3, 1, 1, 5, 10, 10, 5, 2, 5, 10, 10, 5, 1
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row sums are:
{1, 0, 4, 6, 8, 10, 16, 14, 32, 24, 64}.
The problem with Cyclotomic polynomials is there uneven lengths:
Here roots of -1 as (x+1) powers are used to fill out the triangle with positive coefficients.
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FORMULA
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p(x,n)=If[PrimeQ[n], Cyclotomic[n, x]*(x + 1), Cyclotomic[n, x]*(x + 1)^(n + 1 - Length[CoefficientList[Cyclotomic[n, x], x]])]; t(n,m)=Coefficients(p)x,n))
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EXAMPLE
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{1},
{-1, 1},
{1, 2, 1},
{1, 2, 2, 1},
{1, 2, 2, 2, 1},
{1, 2, 2, 2, 2, 1},
{1, 3, 3, 2, 3, 3, 1},
{1, 2, 2, 2, 2, 2, 2, 1},
{1, 4, 6, 4, 2, 4, 6, 4, 1},
{1, 3, 3, 2, 3, 3, 2, 3, 3, 1},
{1, 5, 10, 10, 5, 2, 5, 10, 10, 5, 1}
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MATHEMATICA
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p[x_, n_] := If[PrimeQ[n], Cyclotomic[n, x]*(x + 1), Cyclotomic[n, x]*(x + 1)^(n + 1 - Length[CoefficientList[Cyclotomic[n, x], x]])]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]
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CROSSREFS
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Sequence in context: A154325 A129765 A143187 this_sequence A163994 A156593 A054526
Adjacent sequences: A143206 A143207 A143208 this_sequence A143210 A143211 A143212
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KEYWORD
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uned,probation,sign
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 20 2008
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