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.

A162496 Number of reduced words of length n in the reflection group [3,4,3] of order 1152.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 48, 60, 71, 80, 87, 92, 94, 92, 87, 80, 71, 60, 48, 36, 25, 16, 9, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

Comments

This is also the Weyl group F_4.

References

  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Crossrefs

Cf. A161409, A162493-A162497.

Programs

  • Magma
    G := CoxeterGroup(GrpFPCox, "F4");
f := GrowthFunction(G);
Coefficients(f);

Formula

G.f.: (1-x^2)*(1-x^6)*(1-x^8)*(1-x^12)/(1-x)^4

A162495 Number of reduced words of length n in the icosahedral reflection group [3,5] of order 120.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 12, 12, 12, 12, 11, 9, 7, 5, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

Comments

This group is also the Weyl group H_3.
If the 0's are omitted, this is the coordination sequence for the truncated icosidodecahedron (see Karzes link).
Sometimes "great rhombicosidodecahedron" is preferred when referring in particular to the Archimedean polyhedron with this coordination sequence. - Peter Munn, Mar 22 2021

References

  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
  • David Wells, Archimedean polyhedra in Penguin Dictionary of Curious and Interesting Geometry, Penguin Books, 1991, pp. 6-7.

Links

Crossrefs

Cf. A161409, A162493, A162494, A162496, A162497.

Programs

  • Magma
    G := CoxeterGroup(GrpFPCox, "H3");
f := GrowthFunction(G);
Coefficients(f);

Formula

G.f.: (1-x^2)*(1-x^6)*(1-x^10)/(1-x)^3.

A266783 The growth series for the affine Coxeter (or Weyl) group [3,3,5] (or H_4).

Original entry on oeis.org

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 651, 823, 1024, 1256, 1521, 1821, 2158, 2534, 2952, 3415, 3925, 4485, 5098, 5766, 6491, 7275, 8120, 9028, 10002, 11046, 12162, 13351, 14616, 15960, 17385, 18893, 20486, 22167, 23939, 25805, 27768, 29829, 31989, 34251, 36618, 39092, 41675, 44370, 47180, 50106, 53150, 56315, 59602, 63012
Offset: 0

Views

Author

N. J. A. Sloane, Jan 11 2016

Keywords

References

  • N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, Table 10.
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial; also Table 3.1 page 59.

Links

Crossrefs

For the growth series for the finite group see A162497.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/((1-x)^5*(1-x^11)*(1-x^19)*(1-x^29)) )); // G. C. Greubel, Feb 04 2020
  • Maple
    m:=60; S:=series((1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/((1-x)^5*(1-x^11)*(1-x^19)*(1-x^29)), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 04 2020
  • Mathematica
    CoefficientList[Series[(1 + t)^4 * (1 + t^2) * (1 + t^2 + t^4) * (1 + t^4 + t^8) * (1 + t^2 + t^4 + t^6 + t^8) * (1 + t^6 + t^10 + t^12 + t^16 + t^18 + t^22 + t^24 + t^28 + t^34)/((1 - t) * (1 - t^11) * (1 - t^19) * (1 - t^29)), {t, 0, 60}], t] (* Wesley Ivan Hurt, Apr 12 2017; modified by G. C. Greubel, Feb 04 2020 *)
  • PARI
    Vec( (1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/((1-x)^5*(1-x^11)*(1-x^19)*(1-x^29)) +O('x^60) ) \\ G. C. Greubel, Feb 04 2020
  • Sage
    def A266783_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2)*(1-x^12)*(1-x^20)*(1-x^30)/((1-x)^5*(1-x^11)*(1-x^19)*(1-x^29)) ).list()
A266783_list(60) # G. C. Greubel, Feb 04 2020

Formula

G.f. = t1/t2 where t1 is (1 + t)*(1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + t^10 + t^11)*(1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + t^10 + t^11 + t^12 + t^13 + t^14 + t^15 + t^16 + t^17 + t^18 + t^19)*(1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + t^10 + t^11 + t^12 + t^13 + t^14 + t^15 + t^16 + t^17 + t^18 + t^19 + t^20 + t^21 + t^22 + t^23 + t^24 + t^25 + t^26 + t^27 + t^28 + t^29) and t2 = (1 - t)*(1 - t^11)*(1 - t^19)*(1 - t^29).
G.f.: (1 - x^2)*(1 - x^12)*(1 - x^20)*(1 - x^30)/((1 - x)^5*(1 - x^11)*(1 - x^19)*(1 - x^29)). - G. C. Greubel, Feb 04 2020
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.