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 1-4 of 4 results.

A016283 a(n) = 6^n/8 - 4^(n-1) + 2^(n-3).

Original entry on oeis.org

0, 0, 1, 12, 100, 720, 4816, 30912, 193600, 1194240, 7296256, 44301312, 267904000, 1615810560, 9728413696, 58504691712, 351565004800, 2111537479680, 12677814747136, 76101248090112, 456744927232000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of rectangles that can be formed from the vertices of an n-dimensional cube. E.g., a(3)=12 because the three-dimensional cube has six faces plus six rectangles passing through the center of the cube. Cf. A064436: each rectangle on the cube provides an opportunity for a function not to be a linear threshold function, by alternating in value around the rectangle. - Matthew Cook, Jan 26 2004

Links

Crossrefs

Third column of triangle A075497.
Cf. A025966.

Programs

  • Magma
    [6^n/8 - 4^(n-1) + 2^(n-3): n in [0..25]]; // Vincenzo Librandi, Apr 26 2011
  • Maple
    [seq(9/2*6^n-4*4^n+1/2*2^n,n=0..20)]; # Detlef Pauly (dettodet(AT)yahoo.de), Dec 04 2001
  • Mathematica
    CoefficientList[Series[x^2/((1 - 2 x) (1 - 4 x) (1 - 6 x)), {x, 0, 20}], x] (* Michael De Vlieger, Jan 31 2018 *)
  • Sage
    [((6^n - 2^n)/4-(4^n - 2^n)/2)/2 for n in range(0,21)] # Zerinvary Lajos, Jun 05 2009

Formula

a(n) = (2^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).
G.f.: x^2/((1-2*x)*(1-4*x)*(1-6*x)).
E.g.f.: (exp(2*x) - 8*exp(4*x) + 9*exp(6*x))/2!.
a(n) =((6^n - 2^n)/4 - (4^n - 2^n)/2)/2 , n >= 0. - Zerinvary Lajos, Jun 05 2009

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).

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 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.