A131709 -id:A131709 - cp's OEIS frontend

A089243 Number of partitions into strokes of the star graph with n edges on the plane, up to rotations and reflections around the center node.

1, 2, 3, 4, 9, 22, 61, 200, 689, 3054, 12110, 61132, 274264, 1515134, 7498195, 44301928, 238206692, 1490114770, 8605537805, 56612534420, 348083793872, 2396294898646, 15577794980189, 111781094032984, 763986810923430, 5695585712379834

Offset: 0

Yasutoshi Kohmoto

A "stroke" is defined as follows. If the following conditions are satisfied then the partition to directed paths on a directed graph is called "a partition to strokes on a directed graph", and all directed paths in the partition are called "strokes". C.1. Two different directed paths in a partition do not have the same edges. C.2. A union of two different paths in a partition does not become a directed path. In other words, a "stroke" is a locally maximal path on a directed graph.
This sequence has its origin in the strokes made when writing Japanese Kanji.
The value a(1) is ambiguous as it depends on the definition of the star graph with n = 1 edge. If one of the edge endpoints is labeled as the star center, then we have the current value a(1) = 2. However, if the center is not distinguished, then a(1) would be 1. - Max Alekseyev, May 04 2023

			For n = 3, call the center node "0" and the terminal nodes "1", "2", "3".
Four partitions exist as follows:
  {1->0->2, 0->3}
  {1->0->2, 3->0}
  {1->0, 2->0, 3->0}
  {0->1, 0->2, 0->3}.
So a(3) = 4.

Christian Sievers, Table of n, a(n) for n = 0..728

Cf. A131518, A131520, A354228, A131709, A357857, A357895.

p(n,t,o)=o*sum(k=0,(n-1)/2,n!/(k!*(n-2*k)!)*t^k)+if(n%2==0, n!/(n/2)!*t^(n/2));
a(n)=if(n==0,1,(sumdiv(n,d,eulerphi(n/d)*p(d,n/d,2)) + if(n%2,2*n*p((n-1)/2,2,1),n/2*p(n/2,2,2)+n*p(n/2-1,2,2)+n*p(n/2-1,2,1)))/(2*n)) \\ Christian Sievers, May 14 2023

Edited, terms a(0)-a(1) and a(6) corrected, a(7)-a(13) added by Max Alekseyev, Oct 20 2022

More terms from Christian Sievers, May 14 2023

A134799 a(n) = 3^((3^n - 1)/2).

Original entry on oeis.org

1, 3, 81, 1594323, 12157665459056928801, 5391030899743293631239539488528815119194426882613553319203

Offset: 0

Yasutoshi Kohmoto, Jan 09 2008

Number of partitions into "bus routes" of the graph G_{n+1} defined below.

These seem to be one-third the reduced denominators of Newton's iteration for 1/sqrt(3), starting with 1/3. - Steven Finch, Oct 08 2024

			.........|..................G_1
****
.......__|__................G_2
.........|
****
.__|_____|_____|__..........G_3
...|.....|.....|
.........|
.......__|__
.........|
****.
..._|_........._|_..........G_4
_|__|_____|_____|__|_
.|._|_....|...._|_.|
....|.....|.....|
......_|__|__|_
.......|._|_.|
..........|
****
G_1 = o---. = rooted tree with one edge and one leaf node. For n > 0, G_{n+1} is obtained from G_n by splitting each leaf node into three.

Andrew Howroyd, Table of n, a(n) for n = 0..7
Author?, Mitsumata tree
X. Gourdon and P. Sebah, Pythagoras' Constant.

Cf. A131709, A020760, A376870.

3^((3^Range[0, 6] - 1)/2) (* Paolo Xausa, Oct 17 2024 *)

a(n) is conjectured to be one-third the reduced denominator of b(n) = (3/2)*b(n-1)*(1 - b(n-1)^2); b(0) = 1/3. - Steven Finch, Oct 08 2024

Limit_{n -> oo} A376870(n)/(3*a(n)) = 1/sqrt(3) = A020760. - Steven Finch, Oct 08 2024

Edited by N. J. A. Sloane, Jan 29 2008

a(5) from Andrew Howroyd, Oct 07 2024

A357895 Number of partitions of the complete graph on n vertices into strokes.

Original entry on oeis.org

1, 2, 12, 472, 104800

Offset: 1

Yasutoshi Kohmoto and Max Alekseyev, Oct 18 2022

Partition of graph G=(V,E) into strokes is a collection of directed edge-disjoint trails (viewed as sets of directed edges, cf. A357857) in G such that (i) no two trails can be concatenated into a single one; (ii) the corresponding undirected edges form a partition of E.

Cf. A089243, A131518, A131520, A354228, A131709, A357857.

A135442 Number of partitions into "bus routes" of n X n grid.

Original entry on oeis.org

4, 328

Offset: 1

Yasutoshi Kohmoto, Feb 18 2008

1. One and only one route exists on all edges of G

2. Terminals of two different routes don't meet on the same point

Cf. A131709.

A135444 Number of partitions into "bus routes" of K_n.

Original entry on oeis.org

1, 1, 3, 63

Offset: 1

Yasutoshi Kohmoto, Feb 18 2008

nonn

1. One and only one route exists on all edges of G.

2. Terminals of two different routes don't meet on the same point.

Cf. A131709.

cp's OEIS Frontend

A089243 Number of partitions into strokes of the star graph with n edges on the plane, up to rotations and reflections around the center node.

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A134799 a(n) = 3^((3^n - 1)/2).

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A357895 Number of partitions of the complete graph on n vertices into strokes.

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A135442 Number of partitions into "bus routes" of n X n grid.

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A135444 Number of partitions into "bus routes" of K_n.

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