OFFSET
1,1
COMMENTS
The row sums are all 9.
The octic Algebraic Variety as an Implicit curve:
f[x_, y_, z_] = 9 - 32 x^2 +
32 x^4 - 32 y^2 + 64 x^2 y^2 - 64 x^4 y^2 + 32 y^4 - 64 x^2 y^4 + 64 x^4 y^4 - 32 z^2 + 64 x^2 z^2 - 64 x^4 z^2 + 64 y^2 z^2 - 64 y^4 z^2 + 32 z^4 - 64 x^2 z^4 + 64 x^4z^4 - 64 y^2 z^4 + 64 y^4 z^4 - 1;
has a 24 horn structure with an octahedral shaped ellipsoid embedded.
These quantum states could be expanded to the full
{x,y,z,n,m} to give an analog of a 3d quantum Vafa-Calabi-Yau crystal.
FORMULA
f(x,y,n)=ChebyshevT[n, x]*ChebyshevT[n, y] + ChebyshevT[n, x] + ChebyshevT[n, y]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n); Out_n,m=Coefficients(P(x,1,1,n)).
EXAMPLE
{9},
{5, 4},
{1, 0, 8},
{5, -12, 0, 16},
{9, 0, -32, 0, 32},
{5, 20, 0, -80, 0, 64},
{1, 0, 72, 0, -192, 0, 128},
{5, -28, 0, 224, 0, -448, 0,256},
{9, 0, -128, 0, 640, 0, -1024, 0, 512},
{5, 36,0, -480, 0, 1728, 0, -2304, 0, 1024},
{1, 0, 200, 0, -1600, 0, 4480, 0, -5120, 0, 2048}
MATHEMATICA
f[x_, y_, n_] := ChebyshevT[n, x]*ChebyshevT[n, y] + ChebyshevT[n, x] + ChebyshevT[n, y]; Table[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]], {n, 0, 10}]; a = Table[CoefficientList[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a]
CROSSREFS
KEYWORD
uned,sign
AUTHOR
Roger L. Bagula and Gary W. Adamson, Jun 12 2008
STATUS
approved