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 1-4 of 4 results.

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

A052387 Number of 3 X n binary matrices such that any 2 rows have a common 1, up to column permutations.

Original entry on oeis.org

0, 1, 8, 37, 127, 358, 876, 1926, 3894, 7359, 13156, 22451, 36829, 58396, 89896, 134844, 197676, 283917, 400368, 555313, 758747, 1022626, 1361140, 1791010, 2331810, 3006315, 3840876, 4865823, 6115897, 7630712, 9455248
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Mar 11 2000

Keywords

Links

Crossrefs

Cf. A051588, A051587, A051589.

Programs

  • Magma
    [n*(n+1)*(n+2)*(n+3)*(n^3+22*n^2+53*n+134)/5040: n in [0..30]]; // Wesley Ivan Hurt, May 15 2014
  • Maple
    A052387:=n->n*(n+1)*(n+2)*(n+3)*(n^3+22*n^2+53*n+134)/5040; seq(A052387(n), n=0..30); # Wesley Ivan Hurt, May 15 2014
  • Mathematica
    Table[n*(n + 1)*(n + 2)*(n + 3)*(n^3 + 22*n^2 + 53*n + 134)/5040, {n,
0, 30}] (* Wesley Ivan Hurt, May 15 2014 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(-x*(x^3-x^2-1)/(x-1)^8)) \\ G. C. Greubel, Oct 07 2017

Formula

a(n) = n*(n+1)*(n+2)*(n+3)*(n^3 +22*n^2 +53*n +134)/5040.
G.f.: -x*(x^3-x^2-1)/(x-1)^8. - Colin Barker, Nov 05 2012

A052388 Number of 4 X n binary matrices such that any 2 rows have a common 1, up to column permutations.

Original entry on oeis.org

0, 1, 16, 146, 955, 4905, 20907, 76851, 250530, 739612, 2009177, 5085119, 12109526, 27348478, 58955082, 121956402, 243172488, 469115187, 878387366, 1600751976, 2845918041, 4946262815, 8419256605, 14057377245, 23055913530, 37192403430, 59075703351, 92488040301
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Mar 11 2000

Keywords

References

  • V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Links

Crossrefs

Cf. A051588, A051587, A051589.

Programs

  • Magma
    [n*(n+1)*(n+2)*(n+3)*(n+4)*(n^10 +110*n^9 +5445*n^8 +160050*n^7 +2906463*n^6 +30644250*n^5 +176659055*n^4 +711220750*n^3 +1781493036*n^2 +4034382840*n +4159814400)/1307674368000: n in [0..25]]; // G. C. Greubel, Oct 07 2017
  • Mathematica
    CoefficientList[Series[-x*(x^10 -5*x^9 +10*x^8 -14*x^7 +21*x^6 -19*x^5 -5*x^4 +21*x^3 -10*x^2 -1)/(x-1)^16, {x, 0, 50}], x] (* G. C. Greubel, Oct 07 2017 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(-x*(x^10 -5*x^9 +10*x^8 -14*x^7 +21*x^6 -19*x^5 -5*x^4 +21*x^3 -10*x^2 -1)/(x-1)^16)) \\ G. C. Greubel, Oct 07 2017

Formula

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n^10 +110*n^9 +5445*n^8 +160050*n^7 +2906463*n^6 +30644250*n^5 +176659055*n^4 +711220750*n^3 +1781493036*n^2 +4034382840*n +4159814400)/1307674368000.
G.f.: -x*(x^10 -5*x^9 +10*x^8 -14*x^7 +21*x^6 -19*x^5 -5*x^4 +21*x^3 -10*x^2 -1)/(x-1)^16. - Colin Barker, Nov 05 2012

A319366 Number of 6 X n binary matrices such that any 2 rows have a common 1.

Original entry on oeis.org

1, 127, 14197, 1527655, 154708741, 14581420567, 1282928605477, 106281575400295, 8370106554738181, 632240233746846007, 46159332156459328357, 3278558540783856976135, 227767526682511220042821, 15545657368091391819871447, 1046175606578621216182684837
Offset: 1

Views

Author

T. V. Raziman, Sep 17 2018

Keywords

Links

Crossrefs

Cf. A051587, A051588, A051589.

Formula

a(n) = 64^n - 15*48^n + 60*40^n - 15*36^n + 30*34^n - 6*33^n - 200*32^n - 180*30^n + 585*28^n + 45*27^n + 60*26^n + 150*25^n - 660*24^n - 360*23^n + 168*22^n - 585*21^n + 1245*20^n + 1665*19^n - 1965*18^n - 2100*17^n + 2170*16^n + 1325*15^n - 1770*14^n - 420*13^n + 1533*12^n - 1105*11^n + 435*10^n - 105*9^n + 15*8^n - 7^n (proved in the Quora answer).
Showing 1-4 of 4 results.

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