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 1-3 of 3 results.

A129025 The first 8 values are predefined, the remaining set to a(n) = 48*prime(n)+n+2.

Original entry on oeis.org

14, 52, 78, 133, 152, 248, 345, 538, 1115, 1404, 1501, 1790, 1983, 2080, 2273, 2562, 2851, 2948, 3237, 3430, 3527, 3816, 4009, 4298, 4683, 4876, 4973, 5166, 5263, 5456, 6129, 6322, 6611, 6708, 7189, 7286, 7575, 7864, 8057, 8346, 8635, 8732, 9213, 9310, 9503
Offset: 0

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Author

Roger L. Bagula, May 06 2007

Keywords

Comments

The motivation for these two sequences is that the order-168 Kleinian n=7 group seems to demand a non-Euclidean E9 type of manifold and my work in cosmology led me to think in terms of an E10 exceptional group.

Crossrefs

Cf. A129024.

Programs

  • Mathematica
    a0 = {14, 52, 78, 133, 152, 248, 345, 538}
a = Table[If[n <= 8, a0[[n]], Prime[n]*48 + n + 2], {n, 1, 25}]
Join[{14,52,78,133,152,248,345,538},Table[48Prime[n]+n+2,{n,9,80}]] (* Harvey P. Dale, Feb 11 2015 *)

Formula

Limit_{n->oo} A129025(n)/A129024(n) = 2.

Extensions

More terms from Harvey P. Dale, Feb 11 2015

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

Cf. A128894, A109161, A129024, A129025.

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

Links

Crossrefs

Cf. A128894, A109161, A129024, A129025.

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 1-3 of 3 results.

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