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.

Previous Showing 11-19 of 19 results.

A007155 (2^(2^n))*(3^(3^n - 2^n)).

Original entry on oeis.org

2, 12, 3888, 297538935552, 675089708540070294583610393745358848, 202089899117162771494026986677587201812698520846319037282893739002965043835199487594696085800714664172178112512
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • J. Berman, Free spectra of 3-element algebras, R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983. (Annotated scanned copy)

Programs

  • Maple
    [seq((2^(2^n))*(3^(3^n - 2^n)),n=0..5)];

A007156 Spectrum of a certain 3-element algebra.

Original entry on oeis.org

3, 11, 197, 129615, 430904428717
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • J. Berman, Free spectra of 3-element algebras, R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983. (Annotated scanned copy)

A302251 The number of nonempty antichains in the lattice of set partitions.

Original entry on oeis.org

1, 2, 9, 346, 79814831
Offset: 1

Views

Author

John Machacek, Apr 04 2018

Keywords

Comments

Computing terms in this sequence is analogous to Dedekind's problem which asks for the number of antichains in the Boolean algebra.
This count excludes the empty antichain consisting of no set partitions.

Examples

			For n = 3 the a(3) = 9 nonempty antichains are:
{1/2/3}
{1/23}
{12/3}
{13/2}
{1/23, 12/3}
{1/23, 13/2}
{12/3, 13/2}
{1/23, 12/3, 13/2}
{123}
Here we have used the usual shorthand notation for set partitions where 1/23 denotes {{1}, {2,3}}.

Links

Crossrefs

Equals A302250 - 1, Cf. A000372, A007153, A003182, A014466.

Programs

  • Sage
    [Posets.SetPartitions(n).antichains().cardinality() - 1 for n in range(4)]
# minus removes the empty antichain

A007157 Essentially n-ary operations in Kleene free algebra.

Original entry on oeis.org

0, 4, 74, 43682, 160297810086
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • J. Berman, Free spectra of 3-element algebras, R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983. (Annotated scanned copy)

Crossrefs

Cf. A007154.

Formula

a(n) = Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * A007154(i). - Sean A. Irvine, Nov 05 2017

Extensions

Title improved by Sean A. Irvine, Nov 05 2017

A007158 Essentially n-ary operations in a certain 3-element algebra.

Original entry on oeis.org

2, 10, 3866, 297538923922, 675089708540070294583609203589639922, 202089899117162771494026986677587201812698520846319037282893739002965043831824039051995734327796615178840634970
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • J. Berman, Free spectra of 3-element algebras, R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983. (Annotated scanned copy)

Crossrefs

Cf. A007155.

Formula

a(n) = Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * A007155(i). - Sean A. Irvine, Nov 05 2017

Extensions

a(5) from Sean A. Irvine, Nov 05 2017

A007159 Essentially n-ary operations in a certain 3-element algebra.

Original entry on oeis.org

3, 8, 178, 129054, 430903911398
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • J. Berman, Free spectra of 3-element algebras, R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983. (Annotated scanned copy)

Crossrefs

Cf. A007156.

Formula

a(n) = Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * A007156(i). - Sean A. Irvine, Nov 05 2017

A007184 a(n) = Product_{k=0..n-1} (2^(2^k - 1) + 1)^C(n,k).

Original entry on oeis.org

1, 2, 18, 39366, 23841243993846402, 81709849111396536877982595092224988258053033320397747014
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • J. Berman, Free spectra of 3-element algebras, R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983. (Annotated scanned copy)

Extensions

Title improved by Sean A. Irvine, Nov 12 2017

A174537 Partial sums of A000372.

Original entry on oeis.org

2, 5, 11, 31, 199, 7780, 7836134, 2414689877132, 56130437231102247784920, 286386577668298411184599588898700746597286
Offset: 0

Views

Author

Jonathan Vos Post, Mar 21 2010

Keywords

Comments

Partial sums of Dedekind numbers. Partial sums of number of monotone Boolean functions of n variables (increasing functions from P(S), the set of subsets of S, to {0,1}). Partial sums of number of antichains of subsets of an n-set. The subsequence of primes in this partial sum begins: 2, 5, 11, 31, 199 is prime (5 in a row, then no more known).

Examples

			a(4) = 2 + 3 + 6 + 20 + 168 = 199 is prime.

Crossrefs

Cf. A000372, A014466, A007153, A003182, A059119.

Formula

a(n) = Sum_{i=0..n} A000372(i) = Sum_{i=0..n} (A014466(i) + 1) = Sum_{i=0..n} (A007153(i) + 2).

Extensions

a(9) from A000372 - Dmitry I. Ignatov, Nov 27 2023

A340319 Decimal expansion of sum of reciprocals of Dedekind numbers.

Original entry on oeis.org

1, 0, 5, 6, 0, 8, 4, 4, 1, 7, 4, 1, 2, 7, 3, 5, 5, 3, 6, 6, 7, 6, 3
Offset: 1

Views

Author

Marco Ripà, Jan 04 2021

Keywords

Comments

The series 1/2 + 1/3 + 1/6 + 1/20 + 1/168 + 1/7581 + 1/7828354 + 1/2414682040998 + 1/56130437228687557907788 + ... converges to 1.0560844174127355366763... (since the number of monotone Boolean functions of n variables is strictly smaller than the number of monotone Boolean functions of (n + 1) variables and the limit of its reciprocal is zero as n approaches infinity).

Examples

			1.0560844174127355366763...

Crossrefs

Cf. A000372, A007153.

Formula

Equals Sum_{k>=0} 1/A000372(k).
Previous Showing 11-19 of 19 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.