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 1-7 of 7 results.

A286847 Array read by antidiagonals: T(m,n) = number of minimal dominating sets in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 2, 6, 2, 4, 7, 7, 4, 4, 18, 16, 18, 4, 7, 39, 53, 53, 39, 7, 9, 75, 154, 306, 154, 75, 9, 13, 155, 436, 1167, 1167, 436, 155, 13, 18, 310, 1268, 4939, 6958, 4939, 1268, 310, 18, 25, 638, 3660, 21313, 40931, 40931, 21313, 3660, 638, 25
Offset: 1

Views

Author

Andrew Howroyd, Aug 01 2017

Keywords

Examples

			Table begins:
===============================================================
m\n|  1   2    3     4       5        6         7          8
---|-----------------------------------------------------------
1  |  1   2    2     4       4        7         9         13...
2  |  2   6    7    18      39       75       155        310...
3  |  2   7   16    53     154      436      1268       3660...
4  |  4  18   53   306    1167     4939     21313      88161...
5  |  4  39  154  1167    6958    40931    254754    1519544...
6  |  7  75  436  4939   40931   349178   3118754   26797630...
7  |  9 155 1268 21313  254754  3118754  40307167  497709474...
8  | 13 310 3660 88161 1519544 26797630 497709474 8863408138...
...

Links

Crossrefs

Rows 1-3 are A253412, A290379, A286848.
Main diagonal is A290382.
Cf. A218354 (dominating sets), A089934 (independent), A286868 (irredundant).
Cf. A286849 (king graph).

A350815 Array read by antidiagonals: T(m,n) is the number of minimum dominating sets in the m X n king graph.

Original entry on oeis.org

1, 2, 2, 1, 4, 1, 4, 2, 2, 4, 3, 16, 1, 16, 3, 1, 12, 4, 4, 12, 1, 8, 4, 3, 256, 3, 4, 8, 4, 64, 1, 144, 144, 1, 64, 4, 1, 32, 8, 16, 79, 16, 8, 32, 1, 13, 8, 4, 4096, 9, 9, 4096, 4, 8, 13, 5, 208, 1, 1024, 1656, 1, 1656, 1024, 1, 208, 5, 1, 80, 13, 64, 408, 64, 64, 408, 64, 13, 80, 1
Offset: 1

Views

Author

Andrew Howroyd, Jan 17 2022

Keywords

Comments

The minimum size of a dominating set is the domination number which in the case of an m X n king graph is given by (ceiling(m/3) * ceiling(n/3)).

Examples

			Table begins:
============================================
m\n | 1  2  3    4    5   6      7     8
----+---------------------------------------
  1 | 1  2  1    4    3   1      8     4 ...
  2 | 2  4  2   16   12   4     64    32 ...
  3 | 1  2  1    4    3   1      8     4 ...
  4 | 4 16  4  256  144  16   4096  1024 ...
  5 | 3 12  3  144   79   9   1656   408 ...
  6 | 1  4  1   16    9   1     64    16 ...
  7 | 8 64  8 4096 1656  64 243856 29744 ...
  8 | 4 32  4 1024  408  16  29744  3600 ...
     ...

Links

Crossrefs

Rows 1..3 are A347633, A350816, A347633.
Main diagonal is A347554.
Cf. A075561, A218663 (dominating sets), A286849 (minimal dominating sets), A303335, A350818, A350819.

Formula

T(n,m) = T(m,n).
T(3*m, 3*n) = 1; T(3*m+1, 3*n) = (m^2 + 5*m + 2)^n; T(3*m+2, 3*n) = (m+2)^n.
T(3*m-1, 3*n-1) = A350819(m, n).

A332390 Array read by antidiagonals: T(m,n) is the number of minimal total dominating sets in the m X n king graph.

Original entry on oeis.org

0, 1, 1, 2, 6, 2, 1, 10, 10, 1, 2, 15, 20, 15, 2, 4, 52, 52, 52, 52, 4, 3, 105, 179, 141, 179, 105, 3, 4, 175, 418, 801, 801, 418, 175, 4, 8, 481, 1167, 2950, 7770, 2950, 1167, 481, 8, 9, 1028, 3498, 9792, 34790, 34790, 9792, 3498, 1028, 9, 10, 2000, 9074, 47527, 184318, 204372, 184318, 47527, 9074, 2000, 10
Offset: 1

Views

Author

Andrew Howroyd, Feb 10 2020

Keywords

Examples

			Array begins:
================================================================
m\n | 1   2    3     4       5        6         7          8
----+-----------------------------------------------------------
  1 | 0   1    2     1       2        4         3          4 ...
  2 | 1   6   10    15      52      105       175        481 ...
  3 | 2  10   20    52     179      418      1167       3498 ...
  4 | 1  15   52   141     801     2950      9792      47527 ...
  5 | 2  52  179   801    7770    34790    184318    1305358 ...
  6 | 4 105  418  2950   34790   204372   1593094   14720683 ...
  7 | 3 175 1167  9792  184318  1593094  16260853  231301551 ...
  8 | 4 481 3498 47527 1305358 14720683 231301551 4570906041 ...
  ...

Links

Crossrefs

Rows 1..4 are A302655, A332392, A332393, A332394.
Main diagonal is A332391.
Cf. A286849, A303114, A303118, A303335, A303378.

Formula

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

A286870 Array read by antidiagonals: T(m,n) = number of irredundant sets in the m X n king graph.

Original entry on oeis.org

2, 3, 3, 5, 5, 5, 9, 11, 11, 9, 15, 25, 43, 25, 15, 26, 51, 133, 133, 51, 26, 44, 113, 463, 647, 463, 113, 44, 76, 235, 1493, 2945, 2945, 1493, 235, 76, 130, 521, 5011, 14217, 22049, 14217, 5011, 521, 130, 223, 1107, 16659, 65627, 147672, 147672, 65627, 16659, 1107, 223
Offset: 1

Views

Author

Andrew Howroyd, Aug 02 2017

Keywords

Examples

			Array begins:
====================================================================
m\n|  1   2     3      4       5         6          7           8
---|----------------------------------------------------------------
1  |  2   3     5      9      15        26         44          76...
2  |  3   5    11     25      51       113        235         521...
3  |  5  11    43    133     463      1493       5011       16659...
4  |  9  25   133    647    2945     14217      65627      322163...
5  | 15  51   463   2945   22049    147672    1043127     7365740...
6  | 26 113  1493  14217  147672   1455385   14656628   151865727...
7  | 44 235  5011  65627 1043127  14656628  218691097  3287831848...
8  | 76 521 16659 322163 7365740 151865727 3287831848 72877697369...
...

Links

Crossrefs

Row 1 is A286887.
Main diagonal is A286871.
Cf. A218663 (dominating sets), A286849 (minimal dominating sets).
Cf. A286868 (grid graph).

A332347 Array read by antidiagonals: T(m,n) is the number of maximal independent sets in the m X n king graph.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 3, 6, 6, 3, 4, 12, 8, 12, 4, 5, 20, 22, 22, 20, 5, 7, 36, 40, 79, 40, 36, 7, 9, 64, 82, 194, 194, 82, 64, 9, 12, 112, 176, 537, 544, 537, 176, 112, 12, 16, 200, 340, 1519, 1882, 1882, 1519, 340, 200, 16, 21, 352, 722, 4011, 6490, 8197, 6490, 4011, 722, 352, 21
Offset: 1

Views

Author

Andrew Howroyd, Feb 10 2020

Keywords

Comments

Also the number of minimal vertex covers in the m X n king graph.

Examples

			Array begins:
=====================================================
m\n | 1   2   3    4     5      6       7       8
----+------------------------------------------------
  1 | 1   2   2    3     4      5       7       9 ...
  2 | 2   4   6   12    20     36      64     112 ...
  3 | 2   6   8   22    40     82     176     340 ...
  4 | 3  12  22   79   194    537    1519    4011 ...
  5 | 4  20  40  194   544   1882    6490   20534 ...
  6 | 5  36  82  537  1882   8197   36301  144409 ...
  7 | 7  64 176 1519  6490  36301  201611 1009321 ...
  8 | 9 112 340 4011 20534 144409 1009321 6214593 ...
  ...

Links

Crossrefs

Rows 1..4 are A000931(n+6), A107383(n+2), A332348, A332349.
Main diagonal is A288956.
Cf. A197054 (grid graph), A218663 (dominating sets), A245013 (independent sets), A286849 (minimal dominating sets).

Formula

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

A286850 Number of minimal dominating sets in the 2 X n king graph.

Original entry on oeis.org

2, 4, 6, 16, 20, 52, 80, 176, 296, 592, 1104, 2064, 3936, 7296, 14048, 25984, 49600, 92736, 175872, 330240, 623232, 1175296, 2213632, 4176128, 7863808, 14838784, 27948544, 52707328, 99320832, 187257856, 352940032, 665276416, 1254090752, 2363805696, 4455927808
Offset: 1

Views

Author

Andrew Howroyd, Aug 01 2017

Keywords

Links

Crossrefs

Row 2 of A286849.

Programs

  • Mathematica
    Table[RootSum[8 - 4 #1^2 - 2 #1^3 - 2 #1^4 + #1^6 &, 36 #1^n - 36 #1^(2 + n) + 55 #1^(3 + n) - 3 #1^(4 + n) + 32 #1^(5 + n) &]/970, {n, 10}] (* Eric W. Weisstein, Aug 04 2017 *)
LinearRecurrence[{0, 2, 2, 4, 0, -8}, {2, 4, 6, 16, 20, 52}, 20] (* Eric W. Weisstein, Aug 03 2017 *)
CoefficientList[Series[-((2 (-1 - 2 x - x^2 - 2 x^3 + 4 x^4 + 4 x^5))/(1 - 2 x^2 - 2 x^3 - 4 x^4 + 8 x^6)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 03 2017 *)
  • PARI
    Vec(2*(1+2*x+x^2+2*x^3-4*x^4-4*x^5)/(1-2*x^2-2*x^3-4*x^4+8*x^6)+O(x^40))

Formula

a(n) = 2*a(n-2)+2*a(n-3)+4*a(n-4)-8*a(n-6) for n>6.
G.f.: 2*x*(1 + 2*x + x^2 + 2*x^3 - 4*x^4 - 4*x^5)/(1 - 2*x^2 - 2*x^3 - 4*x^4 + 8*x^6).

A286881 Number of minimal dominating sets in the n X n king graph.

Original entry on oeis.org

1, 4, 12, 256, 971, 85405, 1997448, 360584008, 34097946429, 16133593980207, 8445394800836595, 9548578220258420637
Offset: 1

Views

Author

Eric W. Weisstein, Aug 02 2017

Keywords

Links

Crossrefs

Main diagonal of A286849.
Cf. A133791 (dominating sets), A286871 (irredundant sets).
Cf. A290382 (grid graph).

Extensions

a(5)-a(9) from Andrew Howroyd, Aug 03 2017
a(10)-a(12) from Christian Sievers, Dec 01 2023
Showing 1-7 of 7 results.

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

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