A140056 Triangle of coefficients: 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

Offset

1,1

Comments

Row sums are all -3.

References

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

Links

Table of n, a(n) for n=1..57.

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);

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,
...

Mathematica

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}];

Crossrefs

Sequence in context: A284993 A292068 A350942 * A240239 A247044 A083663

Adjacent sequences: A140053 A140054 A140055 * A140057 A140058 A140059

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Jun 14 2008

Status

approved