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 1-10 of 11 results. Next

A116986 When convolved with itself and doubled, gives the sequence A000609.

Original entry on oeis.org

1, 1, 3, 23, 443, 23131, 3732309, 2090705825, 4388282355347
Offset: 0

Views

Author

Paul D. Hanna, Apr 02 2006

Keywords

Comments

Do these numbers have any other interpretation? Are they always integers?

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

A002080 Number of N-equivalence classes of self-dual threshold functions of n or fewer variables.

Original entry on oeis.org

1, 2, 4, 12, 81, 1684, 122921, 33207256, 34448225389
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
  • 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. A000609, A002077, A002078.

Formula

a(n) = Sum_{k=1..n} A002077(k)*binomial(n,k) = (1/2^n)*Sum_{k=1..n} A000609(k-1)*binomial(n,k). - Alastair D. King, Mar 17 2023.

Extensions

Better description and corrected value of a(7) from Alastair King (see link) - N. J. A. Sloane, Oct 24 2023

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

A064436 Number of switching functions of n or fewer variables which cannot be realized as threshold gates.

Original entry on oeis.org

0, 0, 2, 152, 63654, 4294872724, 18446744073694523482, 340282366920938463463374607423390140592, 115792089237316195423570985008687907853269984665640564039457583990351590086990
Offset: 0

Views

Author

Labos Elemer, Oct 01 2001

Keywords

Comments

The corresponding systems of linear inequalities are not solvable: linearly non-separable truth or switching functions. Truth functions which ar "non-neurons" and are realizable only as two levels threshold gate networks.

Examples

			n=2: out of the 16 B^2 -> B^1 truth functions, 14 are linearly separable; the 2 exceptions are XOR and its negation: f(x,y) = !xz + x!y and !f(x,y) = xy + !x!y. So a(2)=2. With increasing n, the chance that a switching function belongs to this sequence tends to 1.

Links

Crossrefs

Cf. A000609.

Formula

a(n) = 2^(2^n) - A000609(n).

A065246 Formal neural networks with n components.

Original entry on oeis.org

1, 4, 196, 1124864, 12545225621776, 7565068551396549351877632, 11519413104737198429297238164593057431690816, 3940200619639447921227904010014361380507973927046544666794829340424572177149721061141426654884915640806627990306816
Offset: 0

Views

Author

Labos Elemer, Oct 26 2001

Keywords

Comments

Number of {0,1}^n to {0,1}^n vector-vector maps of which all components are formal neurons (=threshold gates).

Examples

			For n=2 the 14 threshold gates determine 14*14=196 neural nets each built purely from threshold gates. For n=3, 104=A000609(3) formal neurons gives 104^3=a(3) networks, all component functions of which are linearly separable {0,1}^3 -> {0,1} vector-scalar functions.

References

  • Labos E. (1996): Long Cycles and Special Categories of Formal Neuronal Networks. Acta Biologica Hungarica, 47: 261-272.
  • Labos E. and Sette M. (1995): Long Cycle Generation by McCulloch-Pitts Networks(MCP-Nets) with Dense and Sparse Weight Matrices. Proc. of BPTM, McCulloch Memorial Conference [eds:Moreno-Diaz R. and Mira-Mira J., pp. 350-359.], MIT Press, Cambridge,MA,USA.
  • McCulloch, W. S. and Pitts W. (1943): A Logical Calculus Immanent in Nervous Activity. Bull. Math. Biophys. 5:115-133.

Crossrefs

Cf. A000609, A065247, A065248, A064436.

Formula

a(n)=A000609(n)^n; for n>1 a(n) < A057156(n).

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

A065247 Imperfect formal neural networks with n components.

Original entry on oeis.org

0, 0, 60, 15652352, 18446731528483929840, 1461501637330902918203677267647731623106580665344, 3940200619639447921227904010014361380507973
Offset: 0

Views

Author

Labos Elemer, Oct 26 2001

Keywords

Comments

Number of {0,1}^n to {0,1}^n vector-vector maps of which at least one component is not a formal neuron, i.e., some are not threshold gates.

Examples

			For n = 2 the 14 threshold gates determine 14*14 = 196 neural nets each built purely from threshold gates; the remaining 2^(2*4)-14^2 = 256-196 = 60 = a(2) functions are synthesized from both neurons and non-neurons. For n = 3, 104 = A000609(3) formal neurons and 152 non-neurons gives (2^24)-A065246(3) = 15652352 = a(4) nets with at least one linearly non-separable component.

References

  • Labos E. (1996): Long Cycles and Special Categories of Formal Neuronal Networks. Acta Biologica Hungarica, 47: 261-272.
  • Labos E. and Sette M.(1995): Long Cycle Generation by McCulloch-Pitts Networks(MCP-Nets) with Dense and Sparse Weight Matrices. Proc. of BPTM, McCulloch Memorial Conference [eds:Moreno-Diaz R. and Mira-Mira J., pp. 350-359.], MIT Press, Cambridge,MA,USA.
  • McCulloch WS and Pitts W (1943): A Logical Calculus Immanent in Nervous Activity. Bull.Math.Biophys. 5:115-133.

Crossrefs

Cf. A000609, A065246, A065248, A064436.

Formula

a(n)=A057156(n)-A000609(n)^n=A057156(n)-A065246(n).

A065248 Networks with n components.

Original entry on oeis.org

0, 4, 3511808, 16417340254783504656, 1461340738496783113671688672284985566897802138624, 3940200619620187981589093886506105584397793947159777
Offset: 1

Views

Author

Labos Elemer, Oct 26 2001

Keywords

Comments

Number of special {0,1}^n to {0,1}^n vector-vector maps of which all components are non-neurons, i.e. none is a linearly separable switching function.

Examples

			For n=2 XOR and its negation are non-neurons, providing 4 networks, all of which permutations are distinguished from each other. For n=3, 152=A064436(3) switching functions are non-neurons, so 152^3=3511808 networks are constructible without formal neurons as component-functions.

References

  • Labos E. (1996): Long Cycles and Special Categories of Formal Neuronal Networks. Acta Biologica Hungarica, 47: 261-272.
  • Labos E. and Sette M.(1995): Long Cycle Generation by McCulloch-Pitts Networks(MCP-Nets) with Dense and Sparse Weight Matrices. Proc. of BPTM, McCulloch Memorial Conference [eds:Moreno-Diaz R. and Mira-Mira J., pp. 350-359.], MIT Press, Cambridge,MA,USA.
  • McCulloch WS and Pitts W (1943): A Logical Calculus Immanent in Nervous Activity. Bull.Math.Biophys. 5:115-133.

Crossrefs

Cf. A000609, A065246, A065247, A064436.

Formula

a(n)=A064436(n)^n
Showing 1-10 of 11 results. Next

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

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