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A140056
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A triangle of coefficients based on a Pisot like toral binomial model: f(x,y,n)=x^n - y^(n - 1)*x - y^n; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n); the quartic curve from this set of trinomials is Klein's quartic.
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0
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-3, -2, -1, -1, -2, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Row sums are all -3.
This result was unexpected: I was just trying to get
binomials whose toral inverse behaves somewhat like minimal Pisot theta
polynomials at degree 3 and 4. The Klein quartic and related curves coming out
was an added bonus?
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REFERENCES
| Weisstein, Eric W. "Klein Quartic." http://mathworld.wolfram.com/KleinQuartic.html
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FORMULA
| f(x,y,n)=x^n - y^(n - 1)*x - y^n; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n);
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EXAMPLE
| {-3},
{-2, -1},
{-1, -2},
{-1, -1, -1},
{-1, -1, 0, -1},
{-1, -1, 0, 0, -1},
{-1, -1, 0, 0, 0, -1},
{-1, -1, 0, 0, 0, 0, -1},
{-1, -1, 0, 0, 0, 0, 0, -1},
{-1, -1, 0, 0, 0, 0, 0, 0, -1},
{-1, -1, 0, 0, 0, 0, 0, 0, 0, -1}.
Polynomials before lower to x only are:
-3,
-x - y - z,
-xy - x z - y z,
-xy^2 - x^2 z - y z^2,
-x y^3 - x^3 z - y z^3,
-x y^4 - x^4 z - y z^4,
...
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MATHEMATICA
| Clear[p, f, x, y, z, n, m, k, i] (* Pisot binomial of torus type*); (*arranged so that x^3 - x - 1 and x^4 - x^3 - 1 behave as being of this type*); f[x_, y_, n] = If[n > 0, x^n - y^(n - 1)*x - y^n, -1]; p[x_, y_, z_, n_] = f[x, y, n] + f[y, z, n] + f[z, x, n]; Table[ExpandAll[p[x, y, z, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, y, z, n] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[p[x, y, z, n] /. y -> 1 /. z -> 1, x]], {n, 0, 10}];
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CROSSREFS
| Sequence in context: A106689 A027082 A140736 * A083663 A085427 A172130
Adjacent sequences: A140053 A140054 A140055 * A140057 A140058 A140059
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 14 2008
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