A006372 -id:A006372 - cp's OEIS frontend

A019988 Number of ways of embedding a connected graph with n edges in the square lattice.

1, 2, 5, 16, 55, 222, 950, 4265, 19591, 91678, 434005, 2073783, 9979772, 48315186, 235088794, 1148891118, 5636168859, 27743309673

Offset: 1

Russ Cox

It is assumed that all edges have length one. - N. J. A. Sloane, Apr 17 2019
These are referred to as 'polysticks', 'polyedges' or 'polyforms'. - Jack W Grahl, Jul 24 2018
Number of connected subgraphs of the square lattice (or grid) containing n length-one line segments. Configurations differing only a rotation or reflection are not counted as different. The question may also be stated in terms of placing unit toothpicks in a connected arrangement on the square lattice. - N. J. A. Sloane, Apr 17 2019
The solution for n=5 features in the card game Digit. - Paweł Rafał Bieliński, Apr 17 2019

Brian R. Barwell, "Polysticks," Journal of Recreational Mathematics, 22 (1990), 165-175.

D. Goodger, An introduction to Polysticks
M. Keller, Counting polyforms
D. Knuth, Dancing Links, arXiv:cs/0011047 [cs.DS], 2000. (A discussion of backtracking algorithms which mentions some problems of polystick tiling.)
Ed Pegg, Jr., Illustrations of polyforms
N. J. A. Sloane, Illustration of a(1)-a(4)
Eric Weisstein's World of Mathematics, Polyedge
Wikicommons, Polysticks 5-sticks 6-sticks 7-sticks

If only translations (but not rotations) are factored, consider fixed polyedges (A096267).
If reflections are considered different, we obtain the one-sided polysticks, counted by (A151537). - Jack W Grahl, Jul 24 2018
Cf. A001997, A003792, A006372, A059103, A085632, A056841 (tree-like), A348095 (with cycles), A348096 (refined by symmetry), A181528.
See A336281 for another version.
6th row of A366766.

A348095(n) + A056841(n+1) = a(n). - R. J. Mathar, Sep 30 2021

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Feb 20 2002

a(18) from John Mason, Jun 01 2023

A306892 Isomorphism classes of connected 2-regular digraphs on n nodes, allowing multiarcs and loops.

Original entry on oeis.org

1, 1, 2, 5, 14, 50, 265, 1601, 11984, 101884

Offset: 0

R. J. Mathar, Mar 15 2019

The graphs are directed, connected and have indegree=outdegree=2 at each node. Multiarcs (connecting two nodes with the same sense of heading) and loops (edges connecting a node to itself) are permitted.

The sequence of the same family of graphs which are not necessarily connected is A006372 (the Euler transform of this sequence).

			On n=1 node, the graph is the node with two edges looping back to the node.
On n=2 nodes, the graph is either having two pairs of edges (4 edges in total) linking one node to the other, or a loop at each node and two edges (different senses) from one node to the other.

R. J. Mathar, OEIS A306892
R. J. Mathar, 2-regular digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM] (2019).

Cf. A306827 (no loops).

a(8) added by R. J. Mathar, Apr 08 2019

a(9) added by R. J. Mathar, Apr 15 2019

A006373 Numbers of terms in expressions for coefficients of Euler-Lagrange tensors in terms of Riemann-Christoffel curvature tensor and two of its contractions (viz., the Ricci curvature tensor and the Riemann curvature scalar) for n-dimensional differentiable manifolds having a general linear connection.

Original entry on oeis.org

1, 2, 7, 26, 115, 596

Offset: 0

C. C. Briggs (ccb104(AT)vm.cac.psu.edu)

The six known terms of this sequence coincide with the first six terms of A167551. - Johannes W. Meijer, Nov 12 2009

C. C. Briggs, A General Expression for the Quintic Lovelock Tensor, arXiv:gr-qc/9607033, 1996-1997.
C. C. Briggs, A General Expression for the Quartic Lovelock Tensor, arXiv:gr-qc/9703074, 1997.
S. Schreiber & N. J. A. Sloane, Correspondence, 1980

Cf. A006372, A030429.

Cf. A167551.

A045900 Hypothetical numbers of terms in general expressions for coefficients of Lovelock Lagrangians.

Original entry on oeis.org

1, 1, 3, 8, 25, 85, 318, 1234, 4884, 19458, 77727, 310761, 1242853, 4971151, 19884270, 79536639, 318145993, 1272583241, 5090332042, 20361326983, 81445306447, 325781223902, 1303124893253, 5212499570050, 20849998276525, 83399993101506, 333599972400353

Offset: 0

C. C. Briggs (ccb104(AT)psu.edu)

nonn

Hypothetical extension of sequence A006372.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
C. C. Briggs, A General Expression for the Quintic Lovelock Tensor, arXiv:gr-qc/9607033, 1996-1997.
C. C. Briggs, A General Expression for the Quartic Lovelock Tensor, arXiv:gr-qc/9703074, 1997.
C. C. Briggs, Some Possible Features of General Expressions for Lovelock Tensors ..., arXiv:gr-qc/9808050, 1998-2000.

Cf. A000041, A006372.

p = PartitionsP;
a[n_] := a[n] = If[n < 4, {1, 1, 3, 8}[[n+1]], a[n-3] - 3*a[n-2] + 3*a[n-1] - p[n-3] + 3*p[n-2] - 3*p[n-1] + p[n] + 2^(2*(n-3) + 1)];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 24 2019 *)

a(n+3) = 3*a(n+2) - 3*a(n+1) + a(n) + p(n+3) - 3*p(n+2) + 3*p(n+1) - p(n) + 2^(2*n + 1); a(n+4) = 7*a(n+3) - 15*a(n+2) + 13*a(n+1) - 4*a(n) + p(n+4) - 7*p(n+3) + 15*p(n+2) - 13*p(n+1) + 4*p(n); where p(n) is the partition function (i.e. number of partitions of n, A000041) and where p(0) = 1.

a(n) ~ 2^(2*n+1) / 27. - Vaclav Kotesovec, Jul 05 2025

More terms from Jean-François Alcover, Feb 24 2019

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A019988 Number of ways of embedding a connected graph with n edges in the square lattice.

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A306892 Isomorphism classes of connected 2-regular digraphs on n nodes, allowing multiarcs and loops.

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A045900 Hypothetical numbers of terms in general expressions for coefficients of Lovelock Lagrangians.

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