cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A360196 Array read by antidiagonals: T(m,n) is the number of induced cycles in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 5, 14, 24, 14, 5, 6, 20, 58, 58, 20, 6, 7, 27, 125, 229, 125, 27, 7, 8, 35, 251, 749, 749, 251, 35, 8, 9, 44, 490, 2180, 3436, 2180, 490, 44, 9, 10, 54, 948, 6188, 13350, 13350, 6188, 948, 54, 10, 11, 65, 1823, 17912, 50203, 65772, 50203, 17912, 1823, 65, 11
Offset: 2

Views

Author

Andrew Howroyd, Jan 29 2023

Keywords

Comments

Induced cycles are sometimes called chordless cycles (but some definitions require chordless cycles to have a cycle length of at least 4).

Examples

			Array begins:
========================================================
m\n| 2  3   4     5      6       7        8        9 ...
---+----------------------------------------------------
2  | 1  2   3     4      5       6        7        8 ...
3  | 2  5   9    14     20      27       35       44 ...
4  | 3  9  24    58    125     251      490      948 ...
5  | 4 14  58   229    749    2180     6188    17912 ...
6  | 5 20 125   749   3436   13350    50203   196918 ...
7  | 6 27 251  2180  13350   65772   308212  1535427 ...
8  | 7 35 490  6188  50203  308212  1743247 10614143 ...
9  | 8 44 948 17912 196918 1535427 10614143 78586742 ...
   ...

Links

Crossrefs

Main diagonal is A297664.
Rows 2..5 are A000027(n-1), A000096(n-1), A360197, A360198.
Cf. A231829 (undirected cycles), A287151 (connected induced subgraphs), A360199 (induced paths), A360202 (induced trees), A360913 (maximum induced cycles).

Formula

T(m,n) = T(n,m).

A360916 Array read by antidiagonals: T(m,n) is the number of maximum induced paths in the grid graph P_m X P_n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 6, 8, 6, 1, 1, 2, 14, 14, 2, 1, 1, 8, 18, 84, 18, 8, 1, 1, 2, 2, 26, 26, 2, 2, 1, 1, 10, 4, 32, 56, 32, 4, 10, 1, 1, 2, 6, 16, 4, 4, 16, 6, 2, 1, 1, 12, 8, 152, 24, 136, 24, 152, 8, 12, 1, 1, 2, 10, 48, 32, 10, 10, 32, 48, 10, 2, 1
Offset: 1

Views

Author

Andrew Howroyd, Feb 26 2023

Keywords

Comments

A maximum induced path is an induced path of longest length.
T(m,n) is the number of snake-like polyominoes with the maximum possible number of unit squares in an m X n rectangle.

Examples

			Array begins:
========================================
m\n| 1  2  3   4   5   6   7   8   9 ...
---+------------------------------------
1  | 1  1  1   1   1   1   1   1   1 ...
2  | 1  4  2   6   2   8   2  10   2 ...
3  | 1  2  8  14  18   2   4   6   8 ...
4  | 1  6 14  84  26  32  16 152  48 ...
5  | 1  2 18  26  56   4  24  32 108 ...
6  | 1  8  2  32   4 136  10 168  32 ...
7  | 1  2  4  16  24  10  52   4   8 ...
8  | 1 10  6 152  32 168   4 216   8 ...
9  | 1  2  8  48 108  32   8   8  16 ...
  ...

Links

Crossrefs

Main diagonal is A331986.
Cf. A360199, A360913, A360917 (lengths), A360918.

Formula

T(m,n) = T(n,m).

A360918 Array read by antidiagonals: T(m,n) is the number of maximum induced trees in the grid graph P_m X P_n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 10, 10, 10, 1, 1, 4, 26, 26, 4, 1, 1, 24, 2, 32, 2, 24, 1, 1, 8, 10, 50, 50, 10, 8, 1, 1, 56, 50, 56, 22, 56, 50, 56, 1, 1, 16, 194, 64, 608, 608, 64, 194, 16, 1, 1, 128, 4, 72, 182, 4120, 182, 72, 4, 128, 1, 1, 32, 32, 80, 16, 208, 208, 16, 80, 32, 32, 1
Offset: 1

Views

Author

Andrew Howroyd, Feb 26 2023

Keywords

Comments

A maximum induced tree is an induced tree of greatest size.

Examples

			Array begins:
===============================================
m\n| 1  2   3  4   5     6    7     8     9 ...
---+-------------------------------------------
1  | 1  1   1  1   1     1    1     1     1 ...
2  | 1  4   2 10   4    24    8    56    16 ...
3  | 1  2  10 26   2    10   50   194     4 ...
4  | 1 10  26 32  50    56   64    72    80 ...
5  | 1  4   2 50  22   608  182    16     2 ...
6  | 1 24  10 56 608  4120  208  1968 22716 ...
7  | 1  8  50 64 182   208  488   560  1050 ...
8  | 1 56 194 72  16  1968  560 65864 14340 ...
9  | 1 16   4 80   2 22716 1050 14340   166 ...
   ...

Links

Crossrefs

Main diagonal is A360919.
Cf. A360202, A360847, A360913, A360916, A360920 (maximum sizes).

Formula

T(m,n) = T(n,m).

A360915 Array read by antidiagonals: T(m,n) is the length of the longest induced cycle in the grid graph P_m X P_n.

Original entry on oeis.org

4, 4, 4, 4, 8, 4, 4, 10, 10, 4, 4, 12, 12, 12, 4, 4, 14, 14, 14, 14, 4, 4, 16, 16, 16, 16, 16, 4, 4, 18, 20, 18, 18, 20, 18, 4, 4, 20, 22, 24, 20, 24, 22, 20, 4, 4, 22, 24, 26, 28, 28, 26, 24, 22, 4, 4, 24, 28, 28, 30, 32, 30, 28, 28, 24, 4, 4, 26, 30, 32, 32, 36, 36, 32, 32, 30, 26, 4
Offset: 2

Views

Author

Andrew Howroyd, Feb 26 2023

Keywords

Comments

All terms are even since the grid graph is bipartite.

Examples

			Array begins:
==========================================
  m\n| 2  3  4  5  6  7  8  9 10 11 12 ...
-----+------------------------------------
   2 | 4  4  4  4  4  4  4  4  4  4  4 ...
   3 | 4  8 10 12 14 16 18 20 22 24 26 ...
   4 | 4 10 12 14 16 20 22 24 28 30 32 ...
   5 | 4 12 14 16 18 24 26 28 32 36 38 ...
   6 | 4 14 16 18 20 28 30 32 36 42 44 ...
   7 | 4 16 20 24 28 32 36 40 44 48 52 ...
   8 | 4 18 22 26 30 36 40 46 50 56 60 ...
   9 | 4 20 24 28 32 40 46 50 56 62 68 ...
  10 | 4 22 28 32 36 44 50 56 62 70 74 ...
  11 | 4 24 30 36 42 48 56 62 70 76 82 ...
  12 | 4 26 32 38 44 52 60 68 74 82 90 ...
  ...

Links

Crossrefs

Main diagonal is A357357.
Cf. A360196, A360913, A360917, A360920.

Formula

T(m,n) = T(n,m).
T(m,n) = 2*m*n/3 + O(m+n) (Beluhov 2023, Proposition 3). - Pontus von Brömssen, May 08 2023

A360914 Number of maximum induced cycles in the n X n grid graph.

Original entry on oeis.org

0, 1, 1, 7, 90, 1398, 23, 2100, 6840, 2912, 416, 1344, 102198, 1643968, 42592, 48128, 1120128
Offset: 1

Views

Author

Andrew Howroyd, Feb 26 2023

Keywords

Comments

A maximum induced cycle is an induced cycle of longest length.

Examples

			The a(3) = 1 unique cycle of longest length is:
   O O O
   O   O
   O O O
.
The a(4) = 7 maximum induced cycles have length 12 and are the following subgraphs with their rotations and reflections.
   O O O O     O O O     O O O
   O     O   O O   O   O O   O
   O     O   O     O   O   O O
   O O O O   O O O O   O O O

Links

  • Eric Weisstein's World of Mathematics, Grid Graph.

Crossrefs

Main diagonal of A360913.
Cf. A297664, A357357 (lengths).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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