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Search: id:A143374
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| A143374 |
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A new 4 symbol polynomial of the Weaver telegraphic type ( Chebyshev T orthogonal) : dot:ChebyshevT[1, y]; dash:ChebyshevT[3, y]; Letter space: ChebyshevT[2, y] ; Word space: ChebyshevT[4, y] ; p(y)=2 - 3 y - 8 y^2 + 28 y^3 + 32 y^4 - 56 y^5 - 32 y^6 + 32 y^7. |
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+0
1
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| 1, 2, 11, 28, 103, 294, 953, 2840, 8831, 26706, 81913, 248924, 760043, 2313430, 7052901, 21479424, 65450335, 199363994, 607380633, 1850218052, 5636529251, 17170499902, 52307409853, 159344658632, 485416893767, 1478734617762
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The idea is that the polynomials are clearly distinguishable
if they are orthogonal.
Another alternative would be cyclotomic prime based polynomials
or Galois Gf(2^n) maybe.
This method is a "machine based" Morse coding compared to the human "keying".
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REFERENCES
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Claude Shannon and Warren Weaver, A Mathematical Theory of Communication, University of Illinois Press, Chicago, 1963, p37 - 38
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FORMULA
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p(y)=2 - 3 y - 8 y^2 + 28 y^3 + 32 y^4 - 56 y^5 - 32 y^6 + 32 y^7; a(n)=coefficient_expansion(x^13*p(1/x))
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EXAMPLE
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Weaver determinant:
A0 = ChebyshevT[1, y];
B0 = ChebyshevT[3, y];
C0 = ChebyshevT[2, y];
D0 = ChebyshevT[4, y];
Expand[FullSimplify[ExpandAll[y *(-3 + 4 y^2)*(1 - 8 y^2 + 8 y^4)Det[{{-1, (
1/B0 + 1/A0)}, {(1/D0 + 1/C0), 1/A0 + 1/B0 - 1}}]]]].
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MATHEMATICA
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p[y_] =2 - 3 y - 8 y^2 + 28 y^3 + 32 y^4 - 56 y^5 - 32 y^6 + 32 y^7; q[x_] = ExpandAll[x^13*p[1/x]]; a = Table[SeriesCoefficient[Series[1/q[x], {x, 0, 30}], n], {n, 0, 30}]
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CROSSREFS
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Cf. A122762.
Sequence in context: A143651 A054552 A034534 this_sequence A045493 A116038 A136317
Adjacent sequences: A143371 A143372 A143373 this_sequence A143375 A143376 A143377
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KEYWORD
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nonn,uned,probation
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 22 2008
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