A065247 -id:A065247 - cp's OEIS frontend

A065246 Formal neural networks with n components.

1, 4, 196, 1124864, 12545225621776, 7565068551396549351877632, 11519413104737198429297238164593057431690816, 3940200619639447921227904010014361380507973927046544666794829340424572177149721061141426654884915640806627990306816

Offset: 0

Labos Elemer, Oct 26 2001

nonn

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

			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.

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.

Cf. A000609, A065247, A065248, A064436.

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

A065248 Networks with n components.

Original entry on oeis.org

0, 4, 3511808, 16417340254783504656, 1461340738496783113671688672284985566897802138624, 3940200619620187981589093886506105584397793947159777

Offset: 1

Labos Elemer, Oct 26 2001

nonn

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.

			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.

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.

Cf. A000609, A065246, A065247, A064436.

a(n)=A064436(n)^n

cp's OEIS Frontend

A065246 Formal neural networks with n components.

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