A058153 -id:A058153 - cp's OEIS frontend

A058129 Number of nonisomorphic monoids (semigroups with identity) of order n.

0, 1, 2, 7, 35, 228, 2237, 31559, 1668997, 3685886630

Offset: 0

Christian G. Bower, Nov 13 2000

Geoff Cruttwell, Counting Finite Categories, presentation, (2018).
Remigiusz Durka and Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.
Najwa Ghannoum, Investigation of finite categories, Doctoral thesis, Univ. Côte d'Azur (France); Univ. Libanaise (Lebanon), tel-0394832 [math.CT], 2022.
Pierre A. Grillet, Counting Semigroups, Communications in Algebra, 43(2), 574-596, (2014).
Mikhail Kornev, On the Classification of n-Valued Monoids and Groups of Order 3, arXiv:2508.04454 [math.GR], 2025. See p. 11.
Václav Koubek and Vojtěch Rödl, Note on the number of monoids of order n, Commentationes Mathematicae Universitatis Carolinae 026.2 (1985): 309-314.
Eric Postpischil, Posting to sci.math newsgroup, May 21 1990
Clayton Cristiano Silva, Irreducible Numerical Semigroups, University of Campinas, São Paulo, Brazil (2019).
Index entries for sequences related to monoids

Cf. A027851 (number of all nonisomorphic semigroups).

a(n) = 2*A058133(n) - A058132(n).
a(n) < A027851(n) except for equality iff n = 1. - M. F. Hasler, Dec 10 2018
From Elijah Beregovsky, May 13 2025 (Start):
a(n) >= A027851(n-1).
Conjecture: a(n) = A027851(n-1)*(1+o(1)). See Koubek and Rödl paper in the Links.
Conjecture: a(n) = A058153(n)/n! * (1+o(1)). See Grillet paper in the Links. (End)

a(8) from Christian G. Bower, Dec 26 2006
a(0) = 0 prepended by M. F. Hasler, Dec 10 2018
a(9) from Elijah Beregovsky, from the work of G. Cruttwell and R. Leblanc, May 12 2025

A058157 Triangle read by rows: T(n,k) is the number of labeled monoids of order n with k idempotents.

Original entry on oeis.org

1, 2, 2, 3, 18, 12, 16, 180, 288, 140, 30, 2640, 6540, 8380, 3020, 480, 119610, 238200, 421020, 372360, 100362, 840, 25196052, 13786290, 26803000, 36174600, 22822674, 4768624, 22080, 48687313640, 2254725312, 2358499080, 3849768160, 3859581096, 1826525120, 305498328

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
   1;
   2,    2;
   3,   18,   12;
  16,  180,  288,  140;
  30, 2640, 6540, 8380, 3020;
  ...

Index entries for sequences related to monoids

Row sums give A058153.
Column 1: A034383.
Main diagonal is A351731.
Cf. A058137 (isomorphism classes), A058158, A058159 (commutative), A058166.

T(n,k) = A058158(n,k)*n.

a(30)-a(36) from Andrew Howroyd, Feb 15 2022

A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.

Original entry on oeis.org

1, 4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236, 782369683393860737, 6355271576489378132, 51624542295308885793

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,1).

Cf. A002018, A002190, A013192, A028576, A041024, A041027.

Cf. A053120, A058153, A058155, A072754, A075132.

[n le 2 select 4^(n-1) else 8*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 13 2022

LinearRecurrence[{8,1},{1,4},30] (* or *) With[{c=Sqrt[17]},Simplify/@ Table[1/2 (c-4)((c+4)^n-(4-c)^n (33+8c)),{n,30}]] (* Harvey P. Dale, May 07 2012 *)

a[0]:1$ a[1]:4$ a[n]:=8*a[n-1]+a[n-2]$ A088317(n):=a[n]$
makelist(A088317(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

A088317=BinaryRecurrenceSequence(8,1,1,4)
[A088317(n) for n in range(31)] # G. C. Greubel, Dec 13 2022

a(n) = ( (4+sqrt(17))^n + (4-sqrt(17))^n )/2.
a(n) = A086594(n)/2.
Lim_{n -> oo} a(n+1)/a(n) = 4 + sqrt(17).
From Paul Barry, Nov 15 2003: (Start)
E.g.f.: exp(4*x)*cosh(sqrt(17)*x).
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*17^k*4^(n-2*k).
a(n) = (-i)^n * T(n, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)
a(n) = A041024(n-1), n>0. - R. J. Mathar, Sep 11 2008
G.f.: (1-4*x)/(1-8*x-x^2). - Philippe Deléham, Nov 16 2008 and Nov 20 2008
a(n) = (1/2)*((33+8*sqrt(17))*(4-sqrt(17))^(n+2) + (33-8*sqrt(17))*(4+sqrt(17))^(n+2)). - Harvey P. Dale, May 07 2012

A351731 Number of labeled idempotent monoids of order n.

Original entry on oeis.org

1, 2, 12, 140, 3020, 100362, 4768624, 305498328, 25293331098, 2619996058190

Offset: 1

Andrew Howroyd, Feb 17 2022

Main diagonal of A058157.

Cf. A058153, A351730.

a(n) = n * A351731(n-1).

A058154 Number of labeled monoids of order n with a fixed identity.

Original entry on oeis.org

1, 2, 11, 156, 4122, 208672, 18507440, 7892741602

Offset: 1

Christian G. Bower, Nov 14 2000

Index entries for sequences related to monoids

a(n) = A058153(n)/n.

a(8) from Christian G. Bower, Dec 28 2006

A346413 Number of labeled totally ordered monoids with n elements.

Original entry on oeis.org

1, 2, 8, 34, 184, 1218, 9742, 92882, 1053248, 14592054

Offset: 1

Bianca Newell, Jul 15 2021

The terms have been computed using the algorithm described in the referenced paper.

Petr Gajdoš and Martin Kuřil, Ordered semigroups of size at most 7 and linearly ordered semigroups of size at most 10, Semigroup Forum, 89 (2014), 639-663.

Cf. A058153.

cp's OEIS Frontend

A058129 Number of nonisomorphic monoids (semigroups with identity) of order n.

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A058157 Triangle read by rows: T(n,k) is the number of labeled monoids of order n with k idempotents.

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A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.

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A351731 Number of labeled idempotent monoids of order n.

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A058154 Number of labeled monoids of order n with a fixed identity.

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A346413 Number of labeled totally ordered monoids with n elements.

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