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.

A350820 Array read by antidiagonals: T(m,n) is the number of minimum dominating sets in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 1, 6, 1, 4, 3, 3, 4, 3, 12, 10, 12, 3, 1, 2, 29, 29, 2, 1, 8, 17, 1, 2, 1, 17, 8, 4, 2, 2, 52, 52, 2, 2, 4, 1, 20, 11, 92, 22, 92, 11, 20, 1, 13, 2, 46, 2, 13, 13, 2, 46, 2, 13, 5, 24, 1, 4, 3, 288, 3, 4, 1, 24, 5, 1, 2, 3, 324, 344, 34, 34, 344, 324, 3, 2, 1
Offset: 1

Views

Author

Andrew Howroyd, Jan 17 2022

Keywords

Comments

The domination number of the grid graphs is tabulated in A350823.

Examples

			Table begins:
===================================
m\n | 1  2  3  4   5   6  7   8
----+------------------------------
  1 | 1  2  1  4   3   1  8   4 ...
  2 | 2  6  3 12   2  17  2  20 ...
  3 | 1  3 10 29   1   2 11  46 ...
  4 | 4 12 29  2  52  92  2   4 ...
  5 | 3  2  1 52  22  13  3 344 ...
  6 | 1 17  2 92  13 288 34   2 ...
  7 | 8  2 11  2   3  34  2  34 ...
  8 | 4 20 46  4 344   2 34  52 ...
  ...

Links

Crossrefs

Rows 1..4 are A347633, A347558, A350821, A350822.
Main diagonal is A347632.
Cf. A218354 (dominating sets), A286847 (minimal dominating sets), A303293, A350815, A350823.

Formula

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

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).

A350816 Number of minimum dominating sets in the 2 X n king graph.

Original entry on oeis.org

2, 4, 2, 16, 12, 4, 64, 32, 8, 208, 80, 16, 608, 192, 32, 1664, 448, 64, 4352, 1024, 128, 11008, 2304, 256, 27136, 5120, 512, 65536, 11264, 1024, 155648, 24576, 2048, 364544, 53248, 4096, 843776, 114688, 8192, 1933312, 245760, 16384, 4390912, 524288, 32768
Offset: 1

Views

Author

Andrew Howroyd, Jan 17 2022

Keywords

Links

Crossrefs

Row 2 of A350815.
Cf. A000079, A001787, A076616, A347558, A347633.

Programs

  • PARI
    Vec(2*(1 - x)*(1 + 3*x + 4*x^2 + 6*x^3 - 4*x^5 - 8*x^6 - 4*x^7)/(1 - 2*x^3)^3 + O(x^45))
  • PARI
    a(n) = {my(t=n\3); 2^t*if(n%3==0, 1, if(n%3==1, t^2 + 5*t + 2, 2*t + 4))}

Formula

a(n) = 6*a(n-3) - 12*a(n-6) + 8*a(n-9) for n > 9.
G.f.: 2*x*(1 - x)*(1 + 3*x + 4*x^2 + 6*x^3 - 4*x^5 - 8*x^6 - 4*x^7)/(1 - 2*x^3)^3.
a(3*n) = 2^n; a(3*n+1) = (n^2 + 5*n + 2)*2^n; a(3*n+2) = (n + 2)*2^(n+1).
a(3*n) = A000079(n); a(3*n+1) = A076616(n+3); a(3*n+2) = A001787(n+2).

A375566 Array read by antidiagonals: T(m,n) = number of minimum dominating sets in the stacked prism graph C_m X P_n.

Original entry on oeis.org

1, 2, 2, 1, 6, 3, 4, 3, 9, 6, 3, 12, 34, 4, 5, 1, 2, 123, 4, 10, 3, 8, 17, 3, 16, 5, 51, 14, 4, 2, 18, 28, 290, 18, 14, 8, 1, 20, 93, 76, 320, 6, 63, 4, 3, 13, 2, 438, 164, 265, 171, 14, 4, 18, 25, 5, 24, 3, 396, 255, 36, 91, 24, 9, 120, 11, 1, 2, 27, 904, 250, 6, 1526, 60, 2052, 25, 22, 3
Offset: 1

Views

Author

Stephan Mertens, Aug 19 2024

Keywords

Examples

			Table starts:
====================================
m\n |   1   2   3    4    5    6 ...
----+-------------------------------
  1 |   1   2   1    4    3    1 ...
  2 |   2   6   3   12    2   17 ...
  3 |   3   9  34  123    3   18 ...
  4 |   6   4   4   16   28   76 ...
  5 |   5  10   5  290  320  265 ...
 ...

Links

Crossrefs

Main diagonal is A375569.
Rows 1..2 are A347633, A347558.
Column 1 is A347538, column 2 is essentially A347634.
Cf. A286514, A350820, A375603.

A350821 Number of minimum dominating sets in the grid graph P_3 X P_n.

Original entry on oeis.org

1, 3, 10, 29, 1, 2, 11, 46, 1, 3, 12, 60, 1, 4, 16, 78, 1, 5, 21, 103, 1, 6, 27, 134, 1, 7, 34, 172, 1, 8, 42, 218, 1, 9, 51, 273, 1, 10, 61, 338, 1, 11, 72, 414, 1, 12, 84, 502, 1, 13, 97, 603, 1, 14, 111, 718, 1, 15, 126, 848, 1, 16, 142, 994, 1, 17, 159, 1157, 1, 18
Offset: 1

Views

Author

Andrew Howroyd, Jan 17 2022

Keywords

Links

Crossrefs

Row 3 of A350820.
Cf. A347633, A347558.

Formula

a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16) for n > 28.
Showing 1-5 of 5 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.