A022562 -id:A022562 - cp's OEIS frontend

A054920 Number of connected unlabeled reflexive relations with n nodes such that complement is also connected.

2, 4, 68, 2592, 278796, 95720106, 111891292036, 457846756500066, 6664787020904248568, 349363873490889302878250, 66602024342830108271942323060, 46557190064705399729526041154647820

Offset: 1

N. J. A. Sloane, May 24 2000

Andrew Howroyd, Table of n, a(n) for n = 1..50
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A000595, A054919.

A000595 = Import["https://oeis.org/A000595/b000595.txt", "Table"][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
A054919 = Join[{1}, EulerInvTransform[A000595 // Rest]];
a[n_] := 2 A054919[[n + 1]] - A000595[[n + 1]];
Array[a, 50] (* Jean-François Alcover, Sep 01 2019, updated Mar 17 2020 *)

a(n) = 2*A054919(n) - A000595(n).

More terms from Vladeta Jovovic, Jul 17 2000

A118054 Inverse Euler transform of A118052.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 2, 0, 2, 1, 3, 2, -1, 0, -3, 0, -2, 4, 5, 3, 5, -1, 5, 1, 4, 1, -11, -8, -25, -4, -2, 22, 46, 22, 47, -16, 23, -26, -20, -28, -144, -79, -218, 4, 36, 272, 454, 279, 465, -133, 89, -415, -455, -550, -1433, -728, -1676, 455, 1047, 3088, 4750, -7192, 13949, -2177, 8689

Offset: 0

Franklin T. Adams-Watters, Apr 10 2006

sign

Does this sequence remain small? How small?

Cf. A118052.

A118052 = Cases[Import["https://oeis.org/A118052/b118052.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{0}, EulerInvTransform[Rest @ A118052]] (* Jean-François Alcover, Feb 23 2020, updated Mar 17 2020 *)

More terms from A118052 by Jean-François Alcover, Feb 23 2020

A243321 Number of simple connected graphs with n nodes that are bipartite and planar.

Original entry on oeis.org

1, 1, 1, 3, 5, 16, 41, 158, 582, 2749, 13852, 80341, 503582, 3419670, 24533162, 184227017

Offset: 1

Travis Hoppe and Anna Petrone, Jun 03 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 9766535.

Cf. A003216 (bipartite graphs), A003094 (planar graphs).

A164099 = Cases[Import["https://oeis.org/A164099/b164099.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[{0} ~Join~ A164099 + 1] (* Jean-François Alcover, Dec 03 2019, updated Mar 17 2020 *)

Inverse Euler transform of (A164099 + 1). - Falk Hüffner, May 10 2019

a(11)-a(16) added using tinygraph by Falk Hüffner, May 10 2019

A349977 Inverse Euler transform of the classical tribonacci numbers.

Original entry on oeis.org

0, 1, 1, 1, 3, 4, 8, 13, 23, 38, 68, 114, 201, 343, 600, 1037, 1817, 3157, 5543, 9692, 17047, 29952, 52828, 93157, 164743, 291459, 516679, 916626, 1628684, 2896261, 5156925, 9189769, 16393652, 29268223, 52300907, 93529331, 167390342, 299787639, 537281476

Offset: 1

Peter Luschny, Dec 07 2021

nonn

The classical tribonacci numbers are defined a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = 0 and a(1) = a(2) = 1.

See A349904 for the analogous sequence for the shifted tribonacci numbers A000073.

Cf. A349904, A000073.

(* EulerInvTransform is defined in A022562. *)
EulerInvTransform[LinearRecurrence[{1, 1, 1}, {0, 1, 1}, 40]]

# After the Maple program of Alois P. Heinz in A349904.
from functools import cache
from math import comb
def euler_inv_trans(a: callable, len: int):
    @cache
    def h(n: int, k: int):
        if n == 0: return 1
        if k <  1: return 0
        bk = b(k)
        R = range(int(bk == 0), 1 + n // k)
        return sum(comb(bk + j - 1, j) * h(n - k * j, k - 1) for j in R)
    @cache
    def b(n: int): return a(n - 1) - h(n, n - 1)
    return [b(n) for n in range(1, len)]
@cache
def tribonacci(n: int):
    return sum(tribonacci(n - j - 1) for j in range(3)) if n >= 3 else min(n, 1)
print(euler_inv_trans(tribonacci, 40))

A352214 Largest number of maximal claw-free node-induced subgraphs of an n-node graph.

Original entry on oeis.org

1, 1, 1, 4, 7, 11, 23, 44, 71

Offset: 1

Pontus von Brömssen, Mar 08 2022

This sequence is log-superadditive, i.e., a(m+n) >= a(m)*a(n). By Fekete's subadditive lemma, it follows that the limit of a(n)^(1/n) exists and equals the supremum of a(n)^(1/n).

			All graphs with at most three nodes are claw-free, so a(n) = 1 for n <= 3 and any graph is optimal.
For 4 <= n <= 9, the following are all optimal graphs, i.e., graphs that have n nodes and a(n) maximal claw-free subgraphs:
  n = 4: K_{1,3};
  n = 5: K_{1,4};
  n = 6: K_{1,5}, K_{3,3} with one edge removed, and K_{3,3};
  n = 7: K_{3,4} with one edge removed;
  n = 8: K_{4,4} with one edge removed;
  n = 9: K_{4,5} with one edge removed.

Cf. A022562, A086991.

For a list of related sequences, see cross-references in A342211.

a(m+n) >= a(m)*a(n).

Limit_{n->oo} a(n)^(1/n) >= 71^(1/9) = 1.60581... .

A058801 Number of connected vertically indecomposable partial lattices on n unlabeled nodes.

Original entry on oeis.org

1, 2, 6, 25, 116, 625, 3757, 25140, 184511, 1473861, 12711339, 117598686, 1160399052, 12152333659, 134487937252, 1566878426731, 19154490559458

Offset: 2

Christian G. Bower, Dec 28 2000

A partial lattice is a poset where every pair of points has a unique least upper (greatest lower) bound or has no upper (lower) bound.

N. J. A. Sloane, Transforms

Cf. A006966.

A058800 = Cases[Import["https://oeis.org/A058800/b058800.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[Drop[A058800, 3]] // Rest (* Jean-François Alcover, May 10 2019, updated Mar 17 2020 *)

Inverse EULER transform of A058800(n+2).

a(17)-a(18) (computed from A058800) from Jean-François Alcover, May 10 2019

A119559 Inverse Euler transform of A119268.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 6, 12, 24, 49, 108, 239, 554, 1311, 3200

Offset: 0

Franklin T. Adams-Watters, May 30 2006

Is this sequence always nonnegative? If so, is there a combinatorial interpretation?

A119268 = Cases[Import["https://oeis.org/A119268/b119268.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{0}, EulerInvTransform[Rest @ A119268]] (* Jean-François Alcover, Feb 23 2020, updated Mar 17 2020 *)

A296105 a(n) is the number of connected transitive relations over n unlabeled nodes.

Original entry on oeis.org

1, 2, 5, 25, 157, 1325, 14358, 199763, 3549001, 80673244, 2352747542, 88240542454, 4261209044877, 264988507673267, 21207485269909946, 2182146922863398203

Offset: 0

Daniele P. Morelli, Dec 04 2017

Inverse Euler transform of A091073. Here "connected" means that it is possible to reach any vertex starting from any other vertex by traversing edges in some direction, i.e., not necessarily in the direction in which the edges point, as in weakly connected digraphs.

			a(2) = 5 because there are five connected transitive relations up to isomorphism: a->b with no loops, a->b with a loop on a, a->b with a loop on b, a->b->a with no loops, and a->b->a with loops on both a and b.

Cf. A091073 (all unlabeled transitive relations). For the labeled case, see A245731 (connected labeled transitive relations) and A006905 (all labeled transitive relations).

A091073 = Cases[Import["https://oeis.org/A091073/b091073.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[A091073 // Rest] (* Jean-François Alcover, Dec 29 2019, updated Mar 17 2020 *)

cp's OEIS Frontend

A054920 Number of connected unlabeled reflexive relations with n nodes such that complement is also connected.

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A118054 Inverse Euler transform of A118052.

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A243321 Number of simple connected graphs with n nodes that are bipartite and planar.

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A349977 Inverse Euler transform of the classical tribonacci numbers.

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A352214 Largest number of maximal claw-free node-induced subgraphs of an n-node graph.

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A058801 Number of connected vertically indecomposable partial lattices on n unlabeled nodes.

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A119559 Inverse Euler transform of A119268.

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A296105 a(n) is the number of connected transitive relations over n unlabeled nodes.

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