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.

A224766 Number of non-degenerate fanout-free Boolean functions of n variables using And, Or and Not gates.

Original entry on oeis.org

2, 2, 8, 64, 832, 15104, 352256, 10037248, 337936384, 13126565888, 577818263552, 28425821618176, 1545553369366528, 92034646352592896, 5956917762776367104, 416397789920380321792, 31262503202358260924416, 2508985620606225641111552, 214348807882902869374926848
Offset: 0

Views

Author

N. J. A. Sloane, Apr 30 2013

Keywords

Comments

Apart from initial term and offset, same as A005640, which is the main entry for this sequence.

References

  • J. P. Hayes, Enumeration of fanout-free Boolean functions, J. ACM, 23 (1976), 700-709.

Crossrefs

Programs

  • PARI
    seq(n) = Vec(2*serlaplace(1 - x + serreverse((1 + 2*x - exp(x + O(x*x^n)))/2))) \\ Andrew Howroyd, Mar 28 2025

Formula

a(n) = 2*A005172(n) for n > 0. - Andrew Howroyd, Mar 28 2025

Extensions

Name clarified and a(19) onwards from Andrew Howroyd, Mar 28 2025

A225170 Number of non-degenerate fanout-free Boolean functions of n variables having AND rank 1.

Original entry on oeis.org

2, 4, 32, 416, 7552, 176128, 5018624, 168968192, 6563282944, 288909131776, 14212910809088, 772776684683264, 46017323176296448, 2978458881388183552, 208198894960190160896, 15631251601179130462208, 1254492810303112820555776, 107174403941451434687463424
Offset: 1

Views

Author

N. J. A. Sloane, Apr 30 2013

Keywords

Comments

Apart from initial term, same as A005172, which is the main entry for this sequence.

Crossrefs

Column 1 of A225171.

Programs

  • Mathematica
    max = 16; s = -ProductLog[-Exp[x-1/2]/2] + O[x]^max; Join[{2}, Drop[CoefficientList[s, x]*Range[0, max-1]!, 2]] (* Jean-François Alcover, Oct 18 2016 *)
    a[1] = 2; a[n_] := (Sum[(n + k - 1)!*Sum[(-1)^j/(k - j)!*Sum[(-1)^i*2^(n - i + j - 1)*StirlingS1[n - i + j - 1, j - i]/((n - i + j - 1)!*i!), {i, 0, j}], {j, 1, k}], {k, 1, n - 1}]);
    Array[a, 20] (* Jean-François Alcover, Jun 24 2018, after Vladimir Kruchinin *)
  • PARI
    seq(n) = Vec(serlaplace(serreverse((1 + 2*x - exp(x + O(x*x^n)))/2 ))) \\ Andrew Howroyd, Mar 28 2025

Formula

Hayes (1976, Theorem 3) gives a recurrence.
G.f.: 1/Q(0) + 1, where Q(k)= 1 - 2*x*(k+1) - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
a(n) ~ (log(2)-1/2)^(1/2 - n) * n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 19 2016
a(n) = 2^n * A000311(n). - Andrew Howroyd, Mar 28 2025

A005756 Number of non-degenerate fanout-free Boolean functions of n variables having AND rank 2.

Original entry on oeis.org

4, 24, 304, 5440, 125824, 3566080, 119614464, 4633387008, 203524112384, 9995546722304, 542730361241600, 32281981804347392, 2087454641985945600, 145797871819529650176, 10938609224992417644544, 877346430770422497673216, 74913579745878635293179904, 6784650500440844952024383488
Offset: 2

Views

Author

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Column 2 of A225171.

Formula

Hayes (1976, Theorem 3) gives a recurrence.

Extensions

More terms from Sean A. Irvine, Aug 26 2016
a(18) onwards from Andrew Howroyd, Mar 28 2025

A224767 Number of non-degenerate fanout-free Boolean functions of n variables having AND rank 3.

Original entry on oeis.org

8, 96, 1760, 41280, 1180928, 39875584, 1552320512, 68449628160, 3372052676608, 183551641878528, 10940489276194816, 708687880636596224, 49572421008089939968, 3724036897414481707008, 299029855691955400343552, 25558593192866693643239424, 2316797640852393238300983296
Offset: 3

Views

Author

N. J. A. Sloane, Apr 30 2013

Keywords

References

  • J. P. Hayes, Enumeration of fanout-free Boolean functions, J. ACM, 23 (1976), 700-709.

Crossrefs

Column 3 of A225171.

Formula

Hayes (1976, Theorem 3) gives a recurrence.

Extensions

a(9) onwards from Andrew Howroyd, Mar 28 2025

A224768 Number of non-degenerate fanout-free Boolean functions of n variables having AND rank 4.

Original entry on oeis.org

16, 320, 8000, 237440, 8212736, 325183488, 14520770560, 722332835840, 39624284553216, 2376711326466048, 154762773034827776, 10873462063657123840, 819935630669686767616, 66053636326538996613120, 5661819237475388709928960, 514513306050639489811873792, 49411385426075987313590009856
Offset: 4

Views

Author

N. J. A. Sloane, Apr 30 2013

Keywords

References

  • J. P. Hayes, Enumeration of fanout-free Boolean functions, J. ACM, 23 (1976), 700-709.

Crossrefs

Column 4 of A225171.

Formula

Hayes (1976, Theorem 3) gives a recurrence.

Extensions

a(9) onwards from Andrew Howroyd, Mar 28 2025
Showing 1-5 of 5 results.