cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A129067 Leading term in row n of triangle in A128894.

Original entry on oeis.org

3, 8, 14, 28, 52, 78, 133, 190, 248, 484
Offset: 1

Views

Author

Roger L. Bagula, May 11 2007

Keywords

Comments

The building exceptional group symmetry sequence in Cartan notation is (Deligne-Landsberg): {A1,A2,G2,D4,F4,E6,E7,E7.5,E8,E9} E9 seems to be closer to an E9.5. For a universe which is E8 symmetry to have evolved, there had to be a metastable (explosive) higher energy/ higher temperature state E9.

References

  • J. M. Landsberg, The sextonions and E_{7 1/2} (with L.Manivel) (Advances in Math 201(2006) p143 - 179) page 22

Crossrefs

Programs

  • Mathematica
    (*A128894*) (*http : // www.math.tamu.edu/~jml /: The sextonions and E_{7 1/2} (with L.Manivel) (Advances in Math 201(2006) p143 - 179) : http : // www.math.tamu.edu/~jml/LMsexpub.pdf : page 22*) a = {-4/3, -1, -2/3, 0, 1, 2, 4, 6, 8, 16}; g[a_, k_] := (3*a + 2*k + 5)*Binomial[k + 2*a + 3, k]* Binomial[k + 5*a/2 + 3, k]*Binomial[k + 3*a + 4, k]/((3*a + 5)*Binomial[k + a/2 + 1, k]*Binomial[k + a + 1, k]) b = Table[g[a[[n]], 1], {n, 1, Length[a]}]

Formula

T(a,n) =(3*a + 2*k + 5)*binomial[k + 2*a + 3, k]*binomial[ k + 5*a/2 + 3, k]*binomial[k + 3*a + 4, k]/((3*a + 5)*binomial[k + a/2 + 1, k]*binomial[k + a + 1, k]) b = Table[Table[g[a[[n]], k], {k, 1, n}], {n, 1, Length[a]}]; k=1 T[n,1]

A129068 A128894[n,k] for k=1 : Coxeter numbers as defined by Bulgadaev for exceptional group sequence using critical exponent solution.

Original entry on oeis.org

2, 3, 3, 6, 9, 12, 18, 24, 30, 50
Offset: 1

Views

Author

Roger L. Bagula, May 11 2007

Keywords

Comments

The building exceptional group symmetry sequence in Cartan notation is ( Deligne-Landsberg): {A1,A2,G2,D4,F4,E6,E7,E7.5,E8,E9} The Coxeter number seem to be related to the total powers in the elliptical invariants for exceptional groups. I have used 2/11 for the F4 critical exponent instead of Bulgadaev's 1/4 because 2/11 fits the linearity of the groups better.

References

  • J. M. Landsberg, The sextonions and E_{7 1/2} (with L.Manivel) (Advances in Math 201(2006) p143 - 179) page 22

Crossrefs

Programs

  • Mathematica
    (*S.A Bulgadaev, arXiv : hep - th/9906091v1 12 Jun 1999*)  b = {2/(1 + 3), 2/(2 + 3), 2/5, 1/4, 2/11, 1/7, 1/10, 1/13, 1/16, 1/26}; hg = Flatten[Table[x /. Solve[2/(2 + x) - b[[n]] == 0, x], {n, 1, Length[b]}]]

Formula

Criticalexponent=k/(k+hg)={2/(1 + 3), 2/(2 + 3), 2/5, 1/4, 2/11, 1/7, 1/10, 1/13, 1/16, 1/26}; hg=Coxeter number=(number of roots)/(rank of group) hg = Flatten[Table[x /. Solve[2/(2 + x) - b[[n]] == 0, x], {n, 1, Length[b]}]]
Showing 1-2 of 2 results.