A058155 -id:A058155 - cp's OEIS frontend

A058159 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents.

1, 2, 2, 3, 18, 6, 16, 180, 144, 36, 30, 2040, 3240, 1740, 380, 360, 43170, 81000, 70740, 31680, 6390, 840, 1400112, 2589510, 2976960, 2055480, 832230, 157962, 15360, 110488616, 117733728, 144285960, 130781280, 79626120, 30004128, 5396888, 68040, 30647444544, 9223088112, 8744866704, 8997002280, 7154708400, 4005012816, 1421659512, 243179064

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
    1;
    2,     2;
    3,    18,     6;
   16,   180,   144,    36;
   30,  2040,  3240,  1740,   380;
  360, 43170, 81000, 70740, 31680, 6390;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..55 (rows 1..10)
Index entries for sequences related to monoids

Row sums give A058155.
Column 1: A034382.
Main diagonal: A055512.
Cf. A058142 (isomorphism classes), A058157, A058160.

T(n, k) = A058160(n, k)*n.

Terms a(30) and beyond 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

A058131 Number of isomorphism classes of commutative monoids (commutative semigroups with identity) of order n.

Original entry on oeis.org

0, 1, 2, 5, 19, 78, 421, 2637, 20486, 246458, 11833361

Offset: 0

Christian G. Bower, Nov 13 2000

Remigiusz Durka and Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.
Eric Postpischil, Posting to sci.math newsgroup, May 21 1990
Eric Weisstein's World of Mathematics, Commutative Monoid
Eric Weisstein's World of Mathematics, Additive Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 952.
Index entries for sequences related to monoids

Row sums of A058142.

Cf. A001426, A058129, A058155 (labeled).

a(0) prepended by Jianing Song, Oct 26 2019
a(8) and a(9) from Alex Meiburg, Oct 20 2021
a(10) from Andrew Howroyd, Feb 15 2022

A058156 Number of labeled commutative monoids of order n with a fixed identity.

Original entry on oeis.org

1, 2, 9, 94, 1486, 38890, 1430442, 77291510, 7826336608, 3904851153672

Offset: 1

Christian G. Bower, Nov 14 2000

Index entries for sequences related to monoids

Row sums of A058160.

Cf. A058155.

a(n) = A058155(n)/n.

a(8)-a(9) from Alex Meiburg, Oct 20 2021

a(10) from Andrew Howroyd, Feb 15 2022

cp's OEIS Frontend

A058159 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

SageMath

Formula

A058131 Number of isomorphism classes of commutative monoids (commutative semigroups with identity) of order n.

Views

Author

Keywords

Links

Crossrefs

Extensions

A058156 Number of labeled commutative monoids of order n with a fixed identity.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions