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 1-6 of 6 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).

A360199 Array read by antidiagonals: T(m,n) is the number of induced paths in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 3, 8, 3, 6, 25, 25, 6, 10, 58, 94, 58, 10, 15, 117, 270, 270, 117, 15, 21, 218, 681, 1004, 681, 218, 21, 28, 387, 1597, 3330, 3330, 1597, 387, 28, 36, 666, 3592, 10224, 14864, 10224, 3592, 666, 36, 45, 1123, 7880, 29924, 61165, 61165, 29924, 7880, 1123, 45
Offset: 1

Views

Author

Andrew Howroyd, Jan 29 2023

Keywords

Comments

Paths of length zero are not counted here.

Examples

			Array begins:
============================================================
m\n|  1   2    3     4      5       6        7         8 ...
---+--------------------------------------------------------
1  |  0   1    3     6     10      15       21        28 ...
2  |  1   8   25    58    117     218      387       666 ...
3  |  3  25   94   270    681    1597     3592      7880 ...
4  |  6  58  270  1004   3330   10224    29924     85036 ...
5  | 10 117  681  3330  14864   61165   238897    907148 ...
6  | 15 218 1597 10224  61165  334536  1723535   8647932 ...
7  | 21 387 3592 29924 238897 1723535 11546874  75134416 ...
8  | 28 666 7880 85036 907148 8647932 75134416 629381852 ...
   ...

Links

Crossrefs

Main diagonal is A360200.
Rows 1..2 are A000217(n-1), A360201.
Cf. A287151 (induced connected subgraphs), A288518 (undirected paths), A360196 (induced cycles), A360202 (induced trees), A360916 (maximum induced paths).

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).

A360194 Array read by antidiagonals: T(m,n) is the number of acyclic spanning subgraphs in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 4, 15, 4, 8, 112, 112, 8, 16, 836, 3102, 836, 16, 32, 6240, 85818, 85818, 6240, 32, 64, 46576, 2373870, 8790016, 2373870, 46576, 64, 128, 347648, 65664106, 900013270, 900013270, 65664106, 347648, 128, 256, 2594880, 1816344222, 92146956300, 341008617408, 92146956300, 1816344222, 2594880, 256
Offset: 1

Views

Author

Andrew Howroyd, Jan 29 2023

Keywords

Comments

Acyclic spanning subgraphs are also called spanning forests.

Examples

			Table starts:
========================================================
m\n|  1     2        3           4               5
---+----------------------------------------------------
1  |  1     2        4           8              16 ...
2  |  2    15      112         836            6240 ...
3  |  4   112     3102       85818         2373870 ...
4  |  8   836    85818     8790016       900013270 ...
5  | 16  6240  2373870   900013270    341008617408 ...
6  | 32 46576 65664106 92146956300 129187804977182 ...
   ...

Links

Crossrefs

Rows 1..4 are A000079(n-1), A022026(n-1), A158450, A360195.
Main diagonal is A080691.
Cf. A116469 (spanning trees), A359993 (connected spanning subgraphs), A360202.

A360203 Number of (non-null) induced trees in the n X n grid graph.

Original entry on oeis.org

1, 12, 138, 3568, 277606, 66136452, 48136454388, 106601739449932, 716581962133166734, 14594259085593605592840, 899530518959027898354960664, 167638624754374503965030664785872, 94397539071875018677962029008899452442, 160524233982090828046095750880433748533447560
Offset: 1

Views

Author

Andrew Howroyd, Feb 22 2023

Keywords

Links

Crossrefs

Main diagonal of A360202.
Cf. A059525, A080691, A297664, A360200.

A360846 Array read by antidiagonals: T(m,n) is the number of dominating induced trees in the grid graph P_m X P_n.

Original entry on oeis.org

1, 3, 3, 4, 8, 4, 4, 17, 17, 4, 4, 32, 65, 32, 4, 4, 66, 222, 222, 66, 4, 4, 130, 766, 1280, 766, 130, 4, 4, 262, 2685, 7629, 7629, 2685, 262, 4, 4, 522, 9450, 46032, 78981, 46032, 9450, 522, 4, 4, 1046, 33158, 278419, 820308, 820308, 278419, 33158, 1046, 4
Offset: 1

Views

Author

Andrew Howroyd, Feb 23 2023

Keywords

Comments

A dominating induced tree in a graph is an acyclic connected induced subgraph whose vertices are a dominating set.

Examples

			Table starts:
=======================================================
m\n| 1   2    3      4       5         6          7 ...
---+---------------------------------------------------
1  | 1   3    4      4       4         4          4 ...
2  | 3   8   17     32      66       130        262 ...
3  | 4  17   65    222     766      2685       9450 ...
4  | 4  32  222   1280    7629     46032     278419 ...
5  | 4  66  766   7629   78981    820308    8520021 ...
6  | 4 130 2685  46032  820308  14605388  259809527 ...
7  | 4 262 9450 278419 8520021 259809527 7904828158 ...
  ...

Links

Crossrefs

Main diagonal is A360847.
Rows 1..2 are A113311(n-1), A360848.
Cf. A291872 (connected dominating sets), A360202 (induced trees).

Formula

T(n,m) = T(m,n).
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.