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 1-6 of 6 results.

A003830 Order of universal Chevalley group D_n (3).

Original entry on oeis.org

2, 576, 12130560, 19808719257600, 2579025599882610278400, 27051378802435080953011843891200, 22941271269626791484963824552883153534976000, 1574947942338058195342953134725345263180893951172280320000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Links

Crossrefs

Cf. A003831, A003832, A003834, A003835, A003836, A132037.

Programs

  • Maple
    f:= n -> 3^(n*(n-1))*(3^n-1)*mul(3^(2*k)-1,k=1..n-1):
map(f, [$1..10]); # Robert Israel, Sep 22 2015
  • Mathematica
    f[m_, n_] := m^(n (n - 1)) (m^n - 1) Product[m^(2 k) - 1, {k, n - 1}];
f[3, #] & /@ Range@ 8 (* Michael De Vlieger, Sep 17 2015 *)
  • PARI
    a(n,q=3) = q^(n*(n-1)) * (q^n-1) * prod(k=1,n-1,q^(2*k)-1); \\ Michel Marcus, Sep 17 2015

Formula

a(n) = D(3,n) where D(q,n) = q^(n*(n-1)) * (q^n-1) * Product_{k=1..n-1}(q^(2*k)-1). - Sean A. Irvine, Sep 17 2015
a(n) ~ c * 3^(n*(2*n-1)), where c = A132037. - Amiram Eldar, Jul 07 2025

Extensions

a(8) and formula from Sean A. Irvine, Sep 17 2015

A003831 Order of universal Chevalley group D_n (4).

Original entry on oeis.org

3, 3600, 987033600, 67010895544320000, 1154606796534757164318720000, 5081732431326820541485324550799360000000, 5722569627753465177061732369386833143098255605760000000, 1649493899207759406688161287839326786813727965837588934265143296000000000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Links

Crossrefs

Cf. A003830, A003832, A003834, A003835, A003836.

Programs

  • Maple
    a:= n -> 4^(n*(n-1))*(4^n-1)*mul(4^(2*k)-1, k=1..n-1):
seq(a(n), n=1..8);  # Alois P. Heinz, Jun 24 2025
  • Mathematica
    d[q_, n_] := q^(n*(n-1)) * (q^n-1) * Product[q^(2*k) - 1, {k, 1, n-1}]; Table[d[4, n], {n, 1, 8}] (* Amiram Eldar, Jun 24 2025 *)

Formula

a(n) = D(4,n) where D(q,n) is defined in A003830. - Sean A. Irvine, Sep 17 2015
a(n) ~ c * 4^(n*(2*n-1)), where c = Product_{k>=1} (1 - 1/4^(2*k)) = 0.933594707399... . - Amiram Eldar, Jul 08 2025

A003832 Order of universal Chevalley group D_n (5).

Original entry on oeis.org

4, 14400, 29016000000, 35646156000000000000, 27230655539587500000000000000000, 12987912192212013697265625000000000000000000000, 3870897639595436240585697704315185546875000000000000000000000000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Links

Crossrefs

Cf. A003830, A003831, A003834, A003835, A003836.

Programs

  • Mathematica
    d[q_, n_] := q^(n*(n-1)) * (q^n-1) * Product[q^(2*k) - 1, {k, 1, n-1}]; Table[d[5, n], {n, 1, 8}] (* Amiram Eldar, Jun 24 2025 *)

Formula

a(n) = D(5,n) where D(q,n) is defined in A003830. - Sean A. Irvine, Sep 17 2015
a(n) ~ c * 5^(n*(2*n-1)), where c = Product_{k>=1} (1 - 1/5^(2*k)) = 0.958400102563... . - Amiram Eldar, Jul 08 2025

Extensions

a(7) from Sean A. Irvine, Sep 17 2015

A003834 Order of universal Chevalley group D_n (7).

Original entry on oeis.org

6, 112896, 4635182361600, 450219964711195607040000, 104772288945650279285144527564308480000, 58523113108854003897259712887271278079596939632967680000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Links

Crossrefs

Cf. A003830, A003831, A003832, A003835, A003836.

Programs

  • Mathematica
    d[q_, n_] := q^(n*(n-1)) * (q^n-1) * Product[q^(2*k) - 1, {k, 1, n-1}]; Table[d[7, n], {n, 1, 8}] (* Amiram Eldar, Jun 24 2025 *)

Formula

a(n) = D(7,n) where D(q,n) is defined in A003830. - Sean A. Irvine, Sep 17 2015
a(n) ~ c * 7^(n*(2*n-1)), where c = Product_{k>=1} (1 - 1/7^(2*k)) = 0.979175347148... . - Amiram Eldar, Jul 08 2025

A003836 Order of universal Chevalley group D_n (9).

Original entry on oeis.org

8, 518400, 203039372390400, 516728027484579221176320000, 8618734695485494933763249322971006238720000, 943067434111013598831873524092098584047517678156686295040000000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Links

Crossrefs

Cf. A003830, A003831, A003832, A003834, A003835.

Programs

  • Mathematica
    d[q_, n_] := q^(n*(n-1)) * (q^n-1) * Product[q^(2*k) - 1, {k, 1, n-1}]; Table[d[9, n], {n, 1, 8}] (* Amiram Eldar, Jun 24 2025 *)

Formula

a(n) = D(9,n) where D(q,n) is defined in A003830. - Sean A. Irvine, Sep 17 2015
a(n) ~ c * 9^(n*(2*n-1)), where c = Product_{k>=1} (1 - 1/9^(2*k)) = 0.987501905484... . - Amiram Eldar, Jul 08 2025

Extensions

a(6) from Sean A. Irvine, Sep 17 2015

A180063 Pascal-like triangle with trigonometric properties, row sums = powers of 4; generated from shifted columns of triangle A180062.

Original entry on oeis.org

1, 1, 3, 1, 4, 11, 1, 7, 15, 41, 1, 8, 38, 56, 153, 1, 11, 46, 186, 209, 571, 1, 12, 81, 232, 859, 780, 2131, 1, 15, 93, 499, 1091, 3821, 2911, 7953, 1, 16, 140, 592, 2774, 4912, 16556, 10864, 29681, 1, 19, 156, 1044, 3366, 14418, 21468, 70356, 40545, 110771
Offset: 0

Views

Author

Gary W. Adamson, Aug 08 2010

Keywords

Comments

Row sums = powers of 4, A000302: (1, 4, 16, 64, ...).
Rightmost terms of each row = A001835: (1, 3, 11, 41, 153, 571, ...).
A180063 may be considered N=4 in an infinite set of Pascal-like triangles generated from variants of the Cartan matrix. Such triangles have trigonometric properties in charpolys being the upward sloping diagonals (cf. triangle A180062 = upward sloping diagonals of A180063); as well as row sums = powers of 2,3,4,...
Triangle A125076 = N=3, with row sums powers of 3; (if the original Pascal's triangle A007318 is considered N=2). To generate the infinite set of these Pascal-like triangles we use Cartan matrix variants with (1's in the super and subdiagonals) and (N-1),N,N,N,... as the main diagonal, alternating with (N,N,N,...).
For example, in the current N=4 triangle, row 7 of A180062 relates to the Heptagon and is generated from the 3 X 3 matrix [3,1,0; 1,4,1; 0,1,4], charpoly x^3 - 11x^2 + 38x - 41. Thus row 7 of triangle A180062 = (1, 11, 38, 41) = an upward sloping diagonal of triangle A180063.
The upward sloping diagonals of the infinite set of Pascal-like triangles = denominators in continued fraction convergents to [1,N,1,N,1,N,...] such that Pascal's triangle (N=2, A007318) has the Fibonacci terms generated from [1,1,1,...]. Similarly, for the case (N=3, triangle A125076), the upward sloping diagonals = row terms of triangle A152063 and are denominators in convergents to [1,2,1,2,1,2,...] = (1, 3, 4, 11, 15, ...).
Triangle A180063 is generated from upward sloping diagonals of triangle A180062, sums found as denominators in [1,3,1,3,1,3,...] = (1, 4, 5, 19, ...).

Examples

			First few rows of the triangle:
  1;
  1,  3;
  1,  4,  11;
  1,  7,  15,   41;
  1,  8,  38,   56,  153;
  1, 11,  46,  186,  209,   571;
  1, 12,  81,  232,  859,   780,  2131;
  1, 15,  93,  499, 1091,  3821,  2911,  7953;
  1, 16, 140,  592, 2774,  4912, 16556, 10864, 29681;
  1, 19, 156, 1044, 3366, 14418, 21468, 70356, 40545, 110771;
  ...

Crossrefs

Cf. A007318, A180062, A003835, A000302.

Formula

Given triangle A180062, shift columns upward so that the new triangle A180063 has (n+1) terms per row.
Showing 1-6 of 6 results.

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

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