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.

A360202 Array read by antidiagonals: T(m,n) is the number of (non-null) induced trees in the grid graph P_m X P_n.

Original entry on oeis.org

1, 3, 3, 6, 12, 6, 10, 33, 33, 10, 15, 78, 138, 78, 15, 21, 171, 533, 533, 171, 21, 28, 360, 2003, 3568, 2003, 360, 28, 36, 741, 7453, 23686, 23686, 7453, 741, 36, 45, 1506, 27643, 156614, 277606, 156614, 27643, 1506, 45, 55, 3039, 102432, 1034875, 3234373, 3234373, 1034875, 102432, 3039, 55
Offset: 1

Views

Author

Andrew Howroyd, Feb 22 2023

Keywords

Examples

			Array begins:
=============================================================
m\n|  1   2     3       4        5          6           7 ...
---+---------------------------------------------------------
1  |  1   3     6      10       15         21          28 ...
2  |  3  12    33      78      171        360         741 ...
3  |  6  33   138     533     2003       7453       27643 ...
4  | 10  78   533    3568    23686     156614     1034875 ...
5  | 15 171  2003   23686   277606    3234373    37643572 ...
6  | 21 360  7453  156614  3234373   66136452  1349087217 ...
7  | 28 741 27643 1034875 37643572 1349087217 48136454388 ...
     ...

Links

Crossrefs

Main diagonal is A360203.
Rows 1..2 are A000217, 3*A125128.
Cf. A287151 (connected induced subgraphs), A116469 (spanning trees), A360196 (induced cycles), A360199 (induced paths), A360918 (maximum induced trees).

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).

A360913 Array read by antidiagonals: T(m,n) is the number of maximum induced cycles in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 1, 1, 4, 5, 1, 7, 1, 5, 6, 1, 17, 17, 1, 6, 7, 1, 31, 90, 31, 1, 7, 8, 1, 8, 284, 284, 8, 1, 8, 9, 1, 40, 9, 1398, 9, 40, 1, 9, 10, 1, 120, 163, 10, 10, 163, 120, 1, 10, 11, 1, 8, 1329, 382, 23, 382, 1329, 8, 1, 11, 12, 1, 64, 128, 6098, 190, 190, 6098, 128, 64, 1, 12
Offset: 2

Views

Author

Andrew Howroyd, Feb 26 2023

Keywords

Comments

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

Examples

			Array begins:
=========================================
m\n| 2  3   4    5    6   7    8    9 ...
---+-------------------------------------
2  | 1  2   3    4    5   6    7    8 ...
3  | 2  1   1    1    1   1    1    1 ...
4  | 3  1   7   17   31   8   40  120 ...
5  | 4  1  17   90  284   9  163 1329 ...
6  | 5  1  31  284 1398  10  382 6098 ...
7  | 6  1   8    9   10  23  190  388 ...
8  | 7  1  40  163  382 190 2100   80 ...
9  | 8  1 120 1329 6098 388   80 6840 ...
  ...

Links

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

Crossrefs

Main diagonal is A360914.
Cf. A360196 (induced cycles), A360915 (lengths), A360916, A360918.

Formula

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

A360920 Array read by antidiagonals: T(m,n) is the maximum number of vertices in an induced tree in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 6, 8, 9, 9, 8, 6, 7, 9, 12, 12, 12, 9, 7, 8, 11, 14, 15, 15, 14, 11, 8, 9, 12, 16, 18, 19, 18, 16, 12, 9, 10, 14, 18, 21, 22, 22, 21, 18, 14, 10, 11, 15, 21, 24, 26, 26, 26, 24, 21, 15, 11, 12, 17, 23, 27, 30, 31, 31, 30, 27, 23, 17, 12
Offset: 1

Views

Author

Andrew Howroyd, Feb 26 2023

Keywords

Examples

			Array begins:
===============================================
  m\n|  1  2  3  4  5  6  7  8  9 10 11  12 ...
-----+-----------------------------------------
   1 |  1  2  3  4  5  6  7  8  9 10 11  12 ...
   2 |  2  3  5  6  8  9 11 12 14 15 17  18 ...
   3 |  3  5  7  9 12 14 16 18 21 23 25  27 ...
   4 |  4  6  9 12 15 18 21 24 27 30 33  36 ...
   5 |  5  8 12 15 19 22 26 30 34 37 41  44 ...
   6 |  6  9 14 18 22 26 31 35 39 44 48  52 ...
   7 |  7 11 16 21 26 31 36 41 46 51 56  61 ...
   8 |  8 12 18 24 30 35 41 46 52 58 63  69 ...
   9 |  9 14 21 27 34 39 46 52 59 65 71  78 ...
  10 | 10 15 23 30 37 44 51 58 65 72 79  86 ...
  11 | 11 17 25 33 41 48 56 63 71 79 87  94 ...
  12 | 12 18 27 36 44 52 61 69 78 86 94 102 ...
  ...

Links

Crossrefs

Main diagonal is A360921.
Cf. A360915, A360917, A360918 (maximum induced trees).

Formula

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

A360919 Number of maximum induced trees in the n X n grid graph.

Original entry on oeis.org

1, 4, 10, 32, 22, 4120, 488, 65864, 166, 1296, 872, 37652512, 678412, 1858036816, 337856
Offset: 1

Views

Author

Andrew Howroyd, Feb 26 2023

Keywords

Comments

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

Examples

			The a(3) = 10 maximum induced trees are the following subgraphs and 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
.
The a(4) = 32 maximum induced trees are the following subgraphs and 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
    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 A360918.
Cf. A360203 (induced trees), A360921 (maximum sizes).
Showing 1-5 of 5 results.

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