A000609 Number of threshold functions of n or fewer variables. (Formerly M1285 N0492)

2, 4, 14, 104, 1882, 94572, 15028134, 8378070864, 17561539552946, 144130531453121108

Offset

0,1

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

Table of n, a(n) for n=0..9.

Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.

Nicolle Gruzling, Linear separability of the vertices of an n-dimensional hypercube, M.Sc Thesis, University of Northern British Columbia, 2006. [From W. Lan (wl(AT)fjrtvu.edu.cn), Jun 27 2010]

Samuel C. Gutekunst, Karola Mészáros, and T. Kyle Petersen, Root Cones and the Resonance Arrangement, arXiv:1903.06595 [math.CO], 2019.

Alastair D. King, Comments on A002080 and related sequences based on threshold functions, Mar 17 2023.

Isaac K. Martin, Andrew G. Moore, John T. Daly, Jess J. Meyer, and Teresa M. Ranadive, Design of General Purpose Minimal-Auxiliary Ising Machines, arXiv:2310.16246 [math.OC], 2023. See p. 7.

Chris Mingard, Joar Skalse, Guillermo Valle-Pérez, David Martínez-Rubio, Vladimir Mikulik, and Ard A. Louis, Neural networks are a priori biased towards Boolean functions with low entropy, arXiv:1909.11522 [cs.LG], 2019.

Guido F. Montufar and Jason Morton, When Does a Mixture of Products Contain a Product of Mixtures?, arXiv preprint arXiv:1206.0387 [stat.ML], 2012-2014.

S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971 [Annotated scans of a few pages]

Muroga, Saburo, Iwao Toda, and Satoru Takasu, Theory of majority decision elements, Journal of the Franklin Institute 271.5 (1961): 376-418. [Annotated scans of pages 413 and 414 only]

S. Muroga, T. Tsuboi and C. R. Baugh, Enumeration of threshold functions of eight variables, IEEE Trans. Computers, 19 (1970), 818-825.

S. Muroga, T. Tsuboi and C. R. Baugh, Enumeration of threshold functions of eight variables, IEEE Trans. Computers, 19 (1970), 818-825. [Annotated scanned copy]

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Stephen Wolfram, A New Kind Of Science. page 1102.

Wikipedia, Linear separability [From W. Lan (wl(AT)fjrtvu.edu.cn), Jun 27 2010]

R. O. Winder, Enumeration of seven-argument threshold functions, IEEE Trans. Electron. Computers, 14 (1965), 315-325.

Index entries for "core" sequences

Index entries for sequences related to Boolean functions

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.

Crossrefs

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

Sequence in context: A032052 A005737 A219767 * A245079 A167008 A329234

Adjacent sequences: A000606 A000607 A000608 * A000610 A000611 A000612

Keyword

nonn,hard,core,nice,more

Author

N. J. A. Sloane

Extensions

a(9) from Minfeng Wang, Jun 27 2010

Status

approved