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 1-5 of 5 results.

A053722 Number of n X n binary matrices of order dividing 2 (also number of solutions to X^2=I in GL(n,2)).

Original entry on oeis.org

1, 4, 22, 316, 6976, 373024, 32252032, 6619979776, 2253838544896, 1810098020122624, 2442718932612677632, 7758088894129169760256, 41674675294431186817908736, 526370120583359572695165435904, 11281778621698661853306239290703872
Offset: 1

Views

Author

Vladeta Jovovic, Mar 23 2000

Keywords

Comments

In characteristic 2, A^2 = I if and only if B^2 = 0 where B = I + A, so a(n) is also equal to the number of n X n binary matrices B such that B^2 = 0.
Conjecture: the two matrices I and 0 have the largest number of square roots. Checked for n=1..5. - Alexey Slizkov, Jan 11 2024

References

  • Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053725, A063505, A225371.

Programs

  • Maple
    Q:= Product(1+u/2^i,i=1..infinity)/Product(1-u^2/2^i,i=1..infinity):
S:= series(Q,u,31):
seq(coeff(S,u,n)*mul(2^i-1,i=1..n), n=1..30); # Robert Israel, Mar 26 2018
  • Mathematica
    QP = QPochhammer; Q = (1-x) QP[-x, 1/2]/QP[x^2, 1/2];
Table[(-1)^n QP[2, 2, n] SeriesCoefficient[Q, {x, 0, n}], {n, 1, 14}] (* Jean-François Alcover, Sep 17 2018, from Maple *)
  • SageMath
    g = lambda n: GL(n,2).order() if n>0 else 1
a053722 = lambda n: g(n)*sum(1/(g(k)*g(n-2*k)*2**(k**2+2*k*(n-2*k))) for k in range(1+floor(n/2))) if n>0 else 0
map(a053722, range(25))
# Dmitrii Pasechnik, Oct 02 2015

Formula

a(n) = Sum_{k=0..floor(n/2)} (2^n - 1)(2^{n-1} - 1) ... (2^{n-2k+1}-1) * 2^{k(k-1)/2} / ((2^k - 1)(2^{k-1} - 1) ... (2^1 - 1)). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001 [Corrected using the paper by Morrison, which also mentions that there is an error in this entry. k = 0 contributes 1 to the sum. If omitted, this gives the number of matrices of order exactly 2. Jan Kristian Haugland, Apr 24 2024]

A121231 Number of n X n binary matrices M (that is, real matrices with entries 0 and 1) such that M^2 is also a binary matrix.

Original entry on oeis.org

1, 2, 11, 172, 6327, 474286, 67147431, 17080038508
Offset: 0

Views

Author

Dan Dima, Aug 21 2006

Keywords

Comments

Comments from Brendan McKay, Aug 21 2006: Equivalently, directed graphs (simple but loops allowed) without a few small forbidden subgraphs (those allowing 2 distinct paths of length 2 from vertex x to vertex y for some x,y; I think there are 6 possibilities). One can also consider isomorphism classes of those digraphs.
Comment from Rob Pratt, Aug 03 2008: A121294 provides a lower bound on the maximum number of 1's in such a matrix M. There are cases where a higher number is reached; the following 5 X 5 matrix has 11 ones and its square is binary:
0 0 1 0 0
0 0 0 0 1
1 1 0 0 1
1 1 0 1 0
1 1 0 1 0.
The optimal values seem to match A070214, verified for n <= 7.
Term (5,1) of the n-th power of the 5 X 5 matrix shown is A001045(n), the Jacobsthal sequence. - Gary W. Adamson, Oct 03 2008
a(n) >= A226321(n).

Links

Crossrefs

Cf. A226321, A225371, A055084, A052264, A051589, A069452, A053304, A001045, A121294, A070214.

Extensions

Edited by R. J. Mathar, Oct 01 2008
a(7) from R. H. Hardin, Jun 19 2012. This makes it clear that the old A122527 was really a badly-described version of this sequence, and that a(7) was earlier found by Balakrishnan (bvarada2(AT)jhu.edu), Sep 17 2006. - N. J. A. Sloane, Jun 19 2012
Entry revised by N. J. A. Sloane, Jun 19 2012

A226321 Number of real {0,1} matrices n X n which are the square of a {0,1} matrix.

Original entry on oeis.org

1, 2, 8, 88, 3245, 254411, 37324414
Offset: 0

Views

Author

Giovanni Resta, Jun 03 2013

Keywords

Comments

A121231(n) >= a(n).

Examples

			The 8 solutions when n=2 are:
00 00 00 01 10 10 10 11
00 01 11 01 00 01 10 00
- from _N. J. A. Sloane_, Jun 03 2013

Links

Crossrefs

Cf. A121231, A225371.

A226236 Number of semisimple invertible n X n matrices over GF(2).

Original entry on oeis.org

1, 3, 105, 11025, 5439105, 11193473025, 89273960290305, 2926331900465537025, 380704455834655419367425, 200503685263248842050957082625, 418936006416927720918846481798529025, 3516831926000321799217446305276779638554625
Offset: 1

Views

Author

Victor S. Miller, Jun 04 2013

Keywords

Comments

Fulman shows that the ratio a(n)/A002884(n) converges to Product_{r>=1, r == 0,2,3 mod 5} (1-2^(-(r-1)))/(1-2^(-r)).

Examples

			a(2) = 3, the matrices are [[1,0],[0,1]], [[0,1],[1,1]], [[1,1],[1,0]].

Links

Crossrefs

Cf. A225371 (every semisimple matrix over GF(2) is a square of another matrix).

A274314 a(n) = number of squares in GL(n,2), the ring of invertible n X n matrices over GF(2).

Original entry on oeis.org

1, 1, 3, 126, 11340, 5940840, 12076523928, 95052257647200, 3153668941285723200, 406198470650573931200640, 215366179177149634500004545792, 447870507819487666185959047316144640, 3770394197251690930118967532374966498493440, 126205342254129164806148123600990735262978861434880, 16960349752279776751561660450391351891796348875427924676608
Offset: 0

Views

Author

N. J. A. Sloane, Jun 25 2016

Keywords

Links

Crossrefs

Cf. A266462, A274313, A225371.
Showing 1-5 of 5 results.

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

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