A065247
Imperfect formal neural networks with n components.
Original entry on oeis.org
0, 0, 60, 15652352, 18446731528483929840, 1461501637330902918203677267647731623106580665344, 3940200619639447921227904010014361380507973
Offset: 0
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.
- 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.
A065248
Networks with n components.
Original entry on oeis.org
0, 4, 3511808, 16417340254783504656, 1461340738496783113671688672284985566897802138624, 3940200619620187981589093886506105584397793947159777
Offset: 1
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.
A065426
Incrementally larger terms in the continued fraction (A065645) for the twin prime constant (A005597).
Original entry on oeis.org
0, 1, 16, 18, 21, 405, 1199, 2301, 19965
Offset: 1
-
(* tpc copied from Niklasch reference *)
cof = ContinuedFraction[tpc, 969]; a = -1; k = 1; Do[ While[ cof[[k]] <= a, k++ ]; a = cof[[k]]; Print[a], {n, 1, 9} ]
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\\ Increasing lprec to 30000 gives no further term beyond 19965.
a065246(lprec) = {localprec(lprec); my (m=-1, T=prodeulerrat(1-1/(p-1)^2, 1, 3), c=contfrac(T)); for (k=1, #c, if (c[k]>m, print(c[k],", "); m=c[k]))};
a065246(1000) \\ Hugo Pfoertner, Aug 01 2023
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