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 1-4 of 4 results.

A193768 The domination number of the 4 X n board.

Original entry on oeis.org

2, 3, 4, 4, 6, 7, 7, 8, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67
Offset: 1

Views

Author

Andrew Buchanan, Tanya Khovanova, Alex Ryba, Aug 06 2011

Keywords

Comments

The domination number of a rectangular grid is the minimal number of X-pentominoes or its fragments that can cover the board.

Examples

			You can't cover the 1 by 4 board with an X-pentomino, but you can do it with two of them.

Links

Crossrefs

Row 4 of A350823.
Cf. A193764, A193765, A193766, A193767.

Programs

  • Mathematica
    LinearRecurrence[{2,-1},{2,3,4,4,6,7,7,8,10,10,11},70] (* Harvey P. Dale, Feb 17 2020 *)
  • PARI
    Vec(x*(x^10-2*x^9+x^8+x^7-x^6-x^5+2*x^4-x^3-x+2)/(x-1)^2 + O(x^100)) \\ Colin Barker, Oct 05 2014

Formula

a(n) = n, except for n = 1, 2, 3, 5, 6 or 9. For the exceptions a(n) = n+1.
a(n) = 4n - A193767(n).
a(n) = 2*a(n-1)-a(n-2) for n>11. - Colin Barker, Oct 05 2014
G.f.: x*(x^10-2*x^9+x^8+x^7-x^6-x^5+2*x^4-x^3-x+2) / (x-1)^2. - Colin Barker, Oct 05 2014

A193764 The number of dominoes in a largest saturated domino covering of the n X n board (n>=2).

Original entry on oeis.org

2, 6, 12, 18, 26, 37, 48, 61, 76, 92, 109, 129, 149, 172, 196, 221, 248, 277, 308, 340, 373, 408, 445, 484, 524, 565, 608, 653, 700, 748, 797, 848, 901, 956, 1012, 1069, 1128, 1189, 1252, 1316, 1381, 1448, 1517, 1588, 1660, 1733, 1808, 1885, 1964, 2044, 2125
Offset: 2

Views

Author

Andrew Buchanan, Tanya Khovanova, Alex Ryba, Aug 06 2011

Keywords

Comments

A domino covering of a board is saturated if the removal of any domino leaves an uncovered cell.

Examples

			If you completely cover a 2 X 2 board with 3 dominoes, you can remove one and the board will still be covered. Hence a(2) < 3. On the other hand, you can tile the 2 X 2 board with 2 dominoes and a removal of one of them will leave both cells uncovered. Hence a(2) = 2.

Links

Crossrefs

Cf. A104519, A193765, A193766, A193767, A193768.

Formula

For n > 6, except n = 13, a(n) = n^2 + 4 - floor((n+2)^2/5).
a(n) = n^2 - A104519(n).
Empirical g.f.: x^2*(x^18 -2*x^17 +x^16 -x^13 +2*x^12 -3*x^11 +2*x^10 +x^9 -2*x^8 +2*x^6 -x^5 -2*x^4 -2*x^2 -2*x -2) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Oct 05 2014
Empirical g.f. confirmed with above formula and recurrence in A104519. - Ray Chandler, Jan 25 2024

A193765 The number of dominoes in the largest saturated domino covering of the n X n board plus one (n >= 2).

Original entry on oeis.org

3, 7, 13, 19, 27, 38, 49, 62, 77, 93, 110, 130, 150, 173, 197, 222, 249, 278, 309, 341, 374, 409, 446, 485, 525, 566, 609, 654, 701, 749, 798, 849, 902, 957, 1013, 1070, 1129, 1190, 1253, 1317, 1382, 1449, 1518, 1589, 1661, 1734, 1809, 1886, 1965, 2045, 2126
Offset: 2

Views

Author

Andrew Buchanan, Tanya Khovanova, Alex Ryba, Aug 06 2011

Keywords

Comments

A domino covering of a board is saturated if the removal of any domino leaves an uncovered cell.
In a domino covering of an n X n board, a domino is redundant if its removal leaves a covering of the board. a(n) is the smallest size of board for which any domino covering must include a redundant domino.

Examples

			If you completely cover a 2 X 2 board with 3 dominoes, you can remove one and the board will still be covered. Hence a(2) >= 3. On the other hand, you can tile the 2 by 2 board with 2 dominoes and a removal of one of them will leave both cells uncovered. Hence a(2) = 3.

Links

Crossrefs

Cf. A104519, A193764, A193766, A193767, A193768.

Formula

For n > 6, except n = 13, a(n) = n^2 + 5 - floor((n+2)^2/5).
a(n) = n^2 +1 - A104519(n).
Empirical g.f.: x^2*(x^18 -2*x^17 +x^16 -x^13 +2*x^12 -3*x^11 +2*x^10 +x^9 -2*x^8 +x^6 -2*x^4 -2*x^2 -x -3) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Oct 05 2014
Empirical g.f. confirmed with above formula and recurrence in A104519. - Ray Chandler, Jan 25 2024

A193767 The number of dominoes in a largest saturated domino covering of the 4 by n board.

Original entry on oeis.org

2, 5, 8, 12, 14, 17, 21, 24, 26, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177
Offset: 1

Views

Author

Andrew Buchanan, Tanya Khovanova, Alex Ryba, Aug 06 2011

Keywords

Comments

A domino covering of a board is saturated if the removal of any domino leaves an uncovered cell.

Examples

			You have to have at least two dominoes to cover the 1 by 4 board, each covering the corner. After that anything else you can remove. Hence a(1) = 2.

Links

Crossrefs

Cf. A193764, A193765, A193766, A193768.

Programs

  • Mathematica
    LinearRecurrence[{2,-1},{2,5,8,12,14,17,21,24,26,30,33},60] (* Harvey P. Dale, Mar 25 2025 *)
  • PARI
    Vec(-x*(x^10-2*x^9+x^8+x^7-x^6-x^5+2*x^4-x^3-x-2)/(x-1)^2 + O(x^100)) \\ Colin Barker, Oct 05 2014

Formula

a(n) = 3n, except for n = 1, 2, 3, 5, 6 or 9. For the exceptions a(n) = 3n-1.
a(n) = 4n - A193768(n).
a(n) = 2*a(n-1)-a(n-2) for n>11. - Colin Barker, Oct 05 2014
G.f.: -x*(x^10-2*x^9+x^8+x^7-x^6-x^5+2*x^4-x^3-x-2) / (x-1)^2. - Colin Barker, Oct 05 2014
Showing 1-4 of 4 results.

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