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 1-7 of 7 results.

A002077 Number of N-equivalence classes of self-dual threshold functions of exactly n variables.

Original entry on oeis.org

1, 0, 1, 4, 46, 1322, 112519, 32267168, 34153652752
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 10.
  • S. Muroga and I. Toda, Lower bound on the number of threshold functions, IEEE Trans. Electron. Computers, 17 (1968), 805-806.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002078, A002079, A002080.

Formula

A002080(n) = Sum_{k=1..n} a(k)*binomial(n,k). Also A000609(n-1) = Sum_{k=1..n} a(k)*binomial(n,k)*2^k. - Alastair D. King, Mar 17 2023.

Extensions

Better description from Alastair King, Mar 17 2023

A000609 Number of threshold functions of n or fewer variables.

Original entry on oeis.org

2, 4, 14, 104, 1882, 94572, 15028134, 8378070864, 17561539552946, 144130531453121108
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is also equal to the number of self-dual threshold functions of n+1 or fewer variables. - Alastair D. King, Mar 17 2023.

References

  • Sze-Tsen Hu, Threshold Logic, University of California Press, 1965 see page 57.
  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
  • S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 3.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • C. Stenson, Weighted voting, threshold functions, and zonotopes, in The Mathematics of Decisions, Elections, and Games, Volume 625 of Contemporary Mathematics Editors Karl-Dieter Crisman, Michael A. Jones, American Mathematical Society, 2014, ISBN 0821898663, 9780821898666

Links

Crossrefs

Cf. A000615, A002077-A002080, A109456, A116986.

Formula

a(n) = Sum_{k=0..n} A000615(k)*binomial(n,k) = Sum_{k=0..n} A002079(k)*binomial(n,k)*2^k. Also A002078(n) = (1/2^n)*Sum_{k=0..n} a(k)*binomial(n,k), a(n-1) = Sum_{k=1..n} A002077(k)*binomial(n,k)*2^k, and A002080(n) = (1/2^n)*Sum_{k=1..n} a(k)*binomial(n,k). - Alastair D. King, Mar 17 2023.

Extensions

a(9) from Minfeng Wang, Jun 27 2010

A002079 Number of N-equivalence classes of threshold functions of exactly n variables.

Original entry on oeis.org

2, 1, 2, 9, 96, 2690, 226360, 64646855, 68339572672
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 8.
  • S. Muroga, T. Tsuboi and C. R. Baugh, Enumeration of threshold functions of eight variables, IEEE Trans. Computers, 19 (1970), 818-825.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002077, A002078, A002080.

Formula

A002078(n) = Sum_{k=0..n} a(k)*binomial(n,k). A000609(n) = Sum_{k=0..n} a(k)*binomial(n,k)*2^k. - Alastair D. King, Mar 17 2023.

Extensions

Better description from Alastair King, Mar 17 2023.

A002078 N-equivalence classes of threshold functions of n or fewer variables.

Original entry on oeis.org

2, 3, 6, 20, 150, 3287, 244158, 66291591, 68863243522
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

It appears that this is the BinomialMean transform of A000609. (See A075271 for the definition of the transform.) - John W. Layman, Feb 21 2003. [This is now confirmed by the formulas below. - Alastair D. King, Mar 17 2023.]

References

  • S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 7.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000609, A002077, A002079, A002080, A075271.

Formula

a(n) = Sum_{k=0..n} A002079(k)*binomial(n,k) = (1/2^n)*Sum_{k=0..n} A000609(k)*binomial(n,k). - Alastair D. King, Mar 17 2023

A001532 Number of NP-equivalence classes of self-dual threshold functions of n or fewer variables ; number of majority (i.e., decisive and weighted) games with n players.

Original entry on oeis.org

1, 1, 2, 3, 7, 21, 135, 2470, 175428, 52980624
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • H. M. Gurk and J. R. Isbell. 1959. Simple Solutions. In A. W. Tucker and R. D. Luce (eds.) Contributions to the Theory of Games, Volume 4. Princeton, NJ: Princeton University Press, pp. 247-265. (Case n=6.)
  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
  • S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 23. (Cases until n=9.)
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • J. von Neumann and O. Morgenstern, Theory of games and economic behavior, Princeton University Press, New Jersey, 1944. (Cases n=1 to 5.)

Links

Crossrefs

Cf. A000617, A002077-A002080, A003184, A109456, A132183, A189359.

Formula

a(n) = Sum_{k=1..n} A003184(k). - Alastair D. King, Oct 26 2023

Extensions

a(10) added by W. Lan (wl(AT)fjrtvu.edu.cn), Jun 27 2010
Better description from Alastair King, Mar 17 2023.

A000615 Threshold functions of exactly n variables.

Original entry on oeis.org

2, 2, 8, 72, 1536, 86080, 14487040, 8274797440, 17494930604032
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 4.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N0142 and N0747).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000609.

Formula

A000609(n) = Sum_{k=0..n} a(k)*binomial(n,k). - Alastair D. King, Mar 17 2023.

Extensions

Entry revised by N. J. A. Sloane, Jun 11 2012

A003184 Number of NP-equivalence classes of self-dual threshold functions of exactly n variables.

Original entry on oeis.org

1, 0, 1, 1, 4, 14, 114, 2335, 172958, 52805196
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • H. M. Gurk and J. R. Isbell. 1959. Simple Solutions. In A. W. Tucker and R. D. Luce (eds.) Contributions to the Theory of Games, Volume 4. Princeton, NJ: Princeton University Press, pp. 247-265. Case n=6.
  • S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 24. (Cases n>7.)
  • J. von Neumann and O. Morgenstern, Theory of games and economic behavior, Princeton University Press, New Jersey, 1944. Cases n=1 to 5.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000619, A001532, A002077-A002080.

Formula

a(n) = A001532(n) - A001532(n-1), for n > 1. - Evgeny Luttsev, Sep 09 2014

Extensions

a(9) from Evgeny Luttsev, Sep 09 2014
Better description and new offset from Alastair King, Mar 17 2023
Showing 1-7 of 7 results.

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

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