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 1-10 of 24 results. Next

A053725 Number of n X n binary matrices of order dividing 3 (also number of solutions to X^3=I in GL(n,2)).

Original entry on oeis.org

1, 3, 57, 1233, 75393, 19109889, 6326835201, 6388287561729, 23576681450405889, 120906321631678693377, 1968421511613895105052673, 111055505036706392268074909697, 8965464105556083354144035638870017
Offset: 1

Views

Author

Vladeta Jovovic, Mar 23 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053722, A053846, A053856.
Cf. A053718, A053770, A053771, A053772, A053773, A053774, A053775, A053776, A053777.

Programs

  • PARI
    \\ See Morison theorem 2.6
\\ F(n,q,k) is number of solutions to X^k=I in GL(i, GF(q)) for i=1..n.
\\ q is power of prime and gcd(q, k) = 1.
B(n,q,e)={sum(m=0, n\e, x^(m*e)/prod(k=0, m-1, q^(m*e)-q^(k*e)))}
F(n,q,k)={if(gcd(q,k)<>1, error("no can do")); my(D=ffgen(q)^0); my(f=factor(D*(x^k-1))); my(p=prod(i=1, #f~, (B(n, q, poldegree(f[i,1])) + O(x*x^n))^f[i,2])); my(r=B(n,q,1)); vector(n, i, polcoeff(p, i)/polcoeff(r, i))}
F(10, 2, 3) \\ Andrew Howroyd, Jul 09 2018

A063393 Number of solutions of x^10=1 in general affine group AGL(n,2).

Original entry on oeis.org

2, 10, 92, 23200, 21391520, 35841831040, 95709758320640, 6206883395497062400, 1502803598296957497344000, 654083813715060854940290252800, 450433384822340709737677746549555200
Offset: 1

Views

Author

Vladeta Jovovic, Jul 16 2001

Keywords

Links

Crossrefs

Cf. A063385-A063393, A062710, A053722, A053725, A053718, A053770-A053777.

A063385 Number of solutions of x^2=1 in general affine group AGL(n,2).

Original entry on oeis.org

2, 10, 92, 1696, 59552, 4124800, 556101632, 148425895936, 78099471368192, 81705857229783040, 169694608681978560512, 702657511446831375056896, 5797142351555426979908943872, 95500953266115919784543392890880, 3140561514292519005433439594146168832
Offset: 1

Views

Author

Vladeta Jovovic, Jul 16 2001

Keywords

Links

Crossrefs

Cf. A063386-A063393, A062710, A053722, A053725, A053718, A053770-A053777.

Extensions

More terms from Sean A. Irvine, Apr 23 2023

A053770 Number of n X n binary matrices of order dividing 5 (i.e., number of solutions of X^5=I in GL(n,2)).

Original entry on oeis.org

1, 1, 1, 1345, 666625, 223985665, 65019838465, 105072058957825, 11436238073940148225, 997931868985434228916225, 74706800043914446529756135425, 5321514758546715999509008953114625, 3721818216683598164434468712927276826625
Offset: 1

Views

Author

Vladeta Jovovic, Mar 24 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053722, A053725, A053718.

Programs

Extensions

a(13) from Andrew Howroyd, Jul 09 2018

A053777 Number of n X n binary matrices of order dividing 12 (i.e., number of solutions of X^12=I in GL(n,2)).

Original entry on oeis.org

1, 6, 120, 10368, 2582208, 3143720448, 11692182896640, 219197554267521024, 12804488375721592356864, 3325324798296500862330077184, 2537067900325971750395878897090560, 8900626797123384385697033838119859781632, 65799342288255766009804607851267459830106816512
Offset: 1

Views

Author

Vladeta Jovovic, Mar 24 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053722, A053725, A053718.

Extensions

More terms from Sean A. Irvine, Jan 16 2022

A053846 Number of n X n matrices over GF(3) of order dividing 2 (i.e., number of solutions of X^2=I in GL(n,3)).

Original entry on oeis.org

1, 2, 14, 236, 12692, 1783784, 811523288, 995733306992, 3988947598331024, 43581058503809001248, 1559669026899267564563936, 152805492791495918971070907584, 49094725258525117931062810300451648, 43237014297639482582550110281347475757696, 124920254287369111633119733942816364074145497472
Offset: 0

Views

Author

Vladeta Jovovic, Mar 28 2000

Keywords

Comments

Or, number of n X n invertible diagonalizable matrices over GF(3).

Examples

			a(2) = 14 because we have: {{0, 1}, {1, 0}}, {{0, 2}, {2, 0}}, {{1, 0}, {0, 1}}, {{1, 0}, {0,2}}, {{1, 0}, {1, 2}}, {{1, 0}, {2, 2}}, {{1, 1}, {0, 2}}, {{1,2}, {0, 2}}, {{2, 0}, {0, 1}}, {{2, 0}, {0, 2}}, {{2, 0}, {1,1}}, {{2, 0}, {2, 1}}, {{2, 1}, {0, 1}}, {{2, 2}, {0, 1}}. - _Geoffrey Critzer_, Aug 05 2017

References

  • Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053722, A053290, A290516.
Row sums of A378666.

Programs

  • Maple
    T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, T(n-1, k-1)+3^k*T(n-1, k)))
    end:
a:= n-> add(3^(k*(n-k))*T(n, k), k=0...n):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 06 2017
  • Mathematica
    nn = 14; g[ n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
QFactorial[n, q]] /. q -> 3; G[z_] := Sum[z^k/g[k], {k, 0, nn}];Table[g[n], {n, 0, nn}] CoefficientList[Series[G[z]^2, {z, 0, nn}], z] (* Geoffrey Critzer, Aug 05 2017 *)
  • PARI
    a(n)={my(v=[1]); for(n=1,n,v=vector(#v+1,k,if(k>1, v[k-1]) + if(k<=#v, 3^(k-1)*v[k]))); sum(k=0,n,3^(k*(n-k))*v[k+1])} \\ Andrew Howroyd, Mar 02 2018
  • Python
    from sympy.core.cache import cacheit
@cacheit
def T(n, k): return 0 if k<0 or k>n else 1 if n==0 else T(n - 1, k - 1) + 3**k*T(n - 1, k)
def a(n): return sum(3**(k*(n - k))*T(n, k) for k in range(n + 1))
print([a(n) for n in range(15)]) # Indranil Ghosh, Aug 06 2017, after Maple code

Formula

a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^2. - Geoffrey Critzer, Aug 05 2017

Extensions

More terms from Geoffrey Critzer, Aug 05 2017

A053771 Number of n X n binary matrices of order dividing 6 (i.e., number of solutions of X^6=I in GL(n,2)).

Original entry on oeis.org

1, 6, 78, 6588, 1332288, 1335398688, 2230748717184, 13819713971871744, 219439188546028498944, 16360198814356838801178624, 3333281205541847127897252298752, 2704161270841324410691567986117967872
Offset: 1

Views

Author

Vladeta Jovovic, Mar 24 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053722, A053725, A053718.

A053773 Number of n X n binary matrices of order dividing 8 (i.e., number of solutions of X^8=I in GL(n,2)).

Original entry on oeis.org

1, 1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, 72057594037927936, 1989505896802466922496, 164384949539438492410445824, 47902612878717208996830483841024
Offset: 0

Views

Author

Vladeta Jovovic, Mar 24 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053722, A053725, A053718, A053763.

A053856 Number of n X n matrices over GF(4) of order dividing 2 (i.e., number of solutions of X^2=I in GL(n,4)).

Original entry on oeis.org

1, 16, 316, 69616, 21999616, 74351051776, 374910580965376, 20054412250260176896, 1616592330896738401386496, 1379799701868277127827135922176, 1779219416300608052895431535861170176
Offset: 1

Views

Author

Vladeta Jovovic, Mar 28 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053722.

Extensions

More terms from Sean A. Irvine, Jan 16 2022

A062250 Number of cyclic subgroups of Chevalley group A_n(2) (the group of nonsingular n X n matrices over GF(2) ).

Original entry on oeis.org

1, 5, 79, 6974, 2037136, 2890467344, 14011554132032, 325330342132674560, 27173394819858612320256, 10158190320726534408118452224, 13156630408268153048253765001412608, 80280189722884518774834501142737770774528
Offset: 1

Views

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 01 2001

Keywords

Examples

			a(3) = 1/phi(1)+21/phi(2)+56/phi(3)+42/phi(4)+48/phi(7) = 79.

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053651 (unlabeled case), A053658, A053660, A053718, A053722, A053725, A053771-A053777, A058502, A062552, A062240.

Formula

a(n) = Sum_{d} |{g element of A_n(2): order(g)=d}|/phi(d), where phi=Euler totient function, cf. A000010.

Extensions

More terms from Vladeta Jovovic, Jul 04 2001
Showing 1-10 of 24 results. Next

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

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