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 1-3 of 3 results.

A232567 Number of non-equivalent binary n X n matrices with two nonadjacent 1's.

Original entry on oeis.org

0, 1, 6, 17, 43, 84, 159, 262, 426, 635, 940, 1311, 1821, 2422, 3213, 4124, 5284, 6597, 8226, 10045, 12255, 14696, 17611, 20802, 24558, 28639, 33384, 38507, 44401, 50730, 57945, 65656, 74376, 83657, 94078, 105129, 117459, 130492, 144951, 160190, 177010
Offset: 1

Views

Author

Heinrich Ludwig, Nov 26 2013

Keywords

Comments

Also: Number of non-equivalent ways to place two non-attacking wazirs on an n X n board.
Two matrix elements are considered adjacent if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood).
This sequence counts equivalence classes induced by the dihedral group D_4. If equivalent matrices are distinguished, the number of matrices is A172225(n).

Examples

			There are a(3) = 6 non-equivalent 3 X 3 matrices with two nonadjacent 1's (and no other 1's):
  [1 0 0]    [0 1 0]    [1 0 0]    [0 1 0]    [1 0 1]    [1 0 0]
  |0 0 0|    |0 0 0|    |0 1 0|    |1 0 0|    |0 0 0|    |0 0 1|
  [0 0 1]    [0 1 0]    [0 0 0]    [0 0 0]    [0 0 0]    [0 0 0]

Links

Crossrefs

Cf. A232568, A232569, A239576, A201511, A172225.

Programs

  • PARI
    x='x+O('x^99); concat(0, Vec(x*(1+x+x^2)*(1+3*x-x^2+x^3)/((1+x)^3*(1-x)^5))) \\ Altug Alkan, Mar 14 2018

Formula

a(n) = (n^4 + 2*n^2 - 4*n)/16 if n is even; a(n) = (n^4 + 4*n^2 - 8*n + 3)/16 if n is odd.
G.f.: x * (1 + x + x^2)*(1 + 3*x - x^2 + x^3) / ((1 + x)^3*(1 - x)^5). - Bruno Berselli, Nov 28 2013

A232569 Triangle T(n, k) = number of non-equivalent (mod D_4) binary n X n matrices with k pairwise not adjacent 1's; k=0,...,n^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 1, 3, 6, 6, 3, 1, 0, 0, 0, 0, 1, 3, 17, 40, 62, 45, 20, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 43, 210, 683, 1425, 1936, 1696, 977, 366, 101, 21, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 84, 681, 4015, 16149, 46472, 95838, 143657
Offset: 1

Views

Author

Heinrich Ludwig, Nov 29 2013

Keywords

Comments

Also number of non-equivalent ways to place k non-attacking wazirs on an n X n board.
Two matrix elements are considered adjacent, if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood).
Counted for this sequence are equivalence classes induced by the dihedral group D_4. If equivalent matrices are being destinguished, the corresponding numbers are A232833(n).
Row index starts from n = 1, column index k ranges from 0 to n^2.
T(n, 1) = A008805(n-1); T(n, 2) = A232567(n) for n >= 2; T(n, 3) = A232568(n) for n >= 2;
Into an n X n binary matrix there can be placed maximally A000982(n) = ceiling(n^2/2) pairwise not adjacent 1's.

Examples

			Triangle begins:
1,1;
1,1,1,0,0;
1,3,6,6,3,1,0,0,0,0;
1,3,17,40,62,45,20,4,1,0,0,0,0,0,0,0,0;
1,6,43,210,683,1425,1936,1696,977,366,101,21,5,1,0,0,0,0,0,0,0,0,0,0,0,0;
...
There are T(3, 2) = 6 non-equivalent binary 3 X 3 matrices with 2 not adjacent 1's (and no other 1's):
  [1 0 0]   [0 1 0]   [1 0 0]   [0 1 0]   [1 0 1]   [1 0 0]
  |0 0 0|   |0 0 0|   |0 1 0|   |1 0 0|   |0 0 0|   |0 0 1|
  [0 0 1]   [0 1 0]   [0 0 0]   [0 0 0]   [0 0 0]   [0 0 0]

Links

Crossrefs

Cf. A232567, A232568, A239576, A008805, A000982, A201511, A232833.

A232568 Number of non-equivalent binary n X n matrices with three pairwise nonadjacent 1's.

Original entry on oeis.org

0, 6, 40, 210, 681, 1919, 4443, 9481, 18206, 33164, 56570, 92996, 146175, 223565, 330981, 479779, 678508, 943586, 1287036, 1731654, 2293765, 3004011, 3883935, 4973645, 6300906, 7917064, 9857198, 12185816, 14946491, 18218969, 22056585, 26556551, 31783320
Offset: 2

Views

Author

Heinrich Ludwig, Nov 28 2013

Keywords

Comments

Also: Number of non-equivalent ways to place three non-attacking wazirs on an n X n board.
Two matrix elements are considered adjacent if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood).
Counted for this sequence are equivalence classes induced by the dihedral group D_4. If equivalent matrices are being destinguished, the number of matrices is A172226(n).

Examples

			There are a(3) = 6 non-equivalent 3 X 3 matrices with three pairwise nonadjacent 1's (and no other 1's):
  [1 0 0]    [1 0 1]    [1 0 0]    [1 0 1]   [1 0 1]   [0 1 0]
  |0 1 0|    |0 0 0|    |0 0 1|    |0 0 0|   |0 1 0|   |1 0 1|
  [0 0 1]    [1 0 0]    [0 1 0]    [0 1 0]   [0 0 0]   [0 0 0]

Links

Crossrefs

Cf. A232567, A239576, A232569, A172226.

Programs

  • Maple
    A232568:=n->(n^6-15*n^4+28*n^3+29*n^2-76*n-15-((n+1) mod 2)*(8*n^3-21*n^2+40*n-63))/48; seq(A232568(n), n=2..50); # Wesley Ivan Hurt, Dec 06 2013
  • Mathematica
    Table[(n^6-15n^4+28n^3+29n^2-76n-15-Mod[n+1,2](8n^3-21n^2+40n-63))/48, {n, 2, 50}] (* Wesley Ivan Hurt, Dec 06 2013 *)

Formula

a(n) = (n^6 - 15*n^4 + 20*n^3 + 50*n^2 - 116*n + 48)/48 if n is even; a(n) = (n^6 - 15*n^4 + 28*n^3 + 29*n^2 - 76*n - 15)/48 if n is odd.
G.f.: x^3*(x^9-4*x^8+x^7+12*x^6+9*x^5-70*x^4-77*x^3-84*x^2-22*x-6) / ((x-1)^7*(x+1)^4). - Colin Barker, Dec 06 2013
a(n) = (n^6 - 15n^4 + 28n^3 + 29n^2 - 76n - 15 - ((n+1) mod 2) * (8n^3 - 21n^2 + 40n - 63))/48. - Wesley Ivan Hurt, Dec 06 2013
Showing 1-3 of 3 results.

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