1,1

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?

Weisstein, Eric W. "Klein Quartic." http://mathworld.wolfram.com/KleinQuartic.html

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);

{-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,

...

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}];

Sequence in context: A106689 A027082 A140736 this_sequence A083663 A085427 A083716

Adjacent sequences: A140053 A140054 A140055 this_sequence A140057 A140058 A140059

uned,tabl,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 14 2008