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.

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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).

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).
Showing 1-2 of 2 results.

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