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.

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A376732 Triangle read by rows: T(n,k) is the maximum number of squares covered (i.e., attacked) by k independent (i.e., non-attacking) queens on an n X n chessboard.

Original entry on oeis.org

1, 4, 0, 9, 9, 0, 12, 15, 16, 16, 17, 23, 25, 25, 25, 20, 30, 35, 36, 36, 36, 25, 37, 45, 49, 49, 49, 49, 28, 44, 55, 62, 64, 64, 64, 64, 33, 52, 66, 76, 81, 81, 81, 81, 81, 36, 60, 77, 92, 100, 100, 100, 100, 100, 100, 41, 68, 88, 104, 121, 121, 121, 121, 121, 121, 121
Offset: 1

Views

Author

John King, Oct 03 2024

Keywords

Comments

T(2,2) = T(3,3) = 0 indicate that there are no solutions to the n-queens problem when n is 2 or 3.

Examples

			Triangle begins:
  n\k|  1    2    3    4    5    6    7    8    9   10   11   12
 ----+-----------------------------------------------------------
   1 |  1;
   2 |  4,   0;
   3 |  9,   9,   0;
   4 | 12,  15,  16,  16;
   5 | 17,  23,  25,  25,  25;
   6 | 20,  30,  35,  36,  36,  36;
   7 | 25,  37,  45,  49,  49,  49,  49;
   8 | 28,  44,  55,  62,  64,  64,  64,  64;
   9 | 33,  52,  66,  76,  81,  81,  81,  81,  81;
  10 | 36,  60,  77,  92, 100, 100, 100, 100, 100, 100;
  11 | 41,  68,  88, 104, 121, 121, 121, 121, 121, 121, 121;
  12 | 44,  76, 101, 120, 134, 142, 144, 144, 144, 144, 144, 144;
  13 | 49,  84, 112, 136, 153, 165, 169, 169, 169, 169, 169, ...;
  14 | 52,  92, 125, 152, 172, 186, 194, 196, 196, 196, 196, ...;
  15 | 57, 100, 136, 168, 193, 209, 221, 224, 225, 225, 225, ...;
  16 | 60, 108, 149, 184, 212, 231, 242, 251, 256, 256, 256, ...;
  17 | 65, 116, 160, 200, 233, 255, 269, 281, 289, 289, 289, ...;
  18 | 68, 124, 173, 216, 252, 277, 294, 310, 322, 324, 324, ...;
  ...

Links

Crossrefs

Columns 1..8 are A047461, A374933, A375116, A374934, A374935, A374936, A374937, A374938.
Cf. A075324, A002567, A075458, A274947.

Formula

T(n,k) = n^2 for k >= A075324(n), n >= 4.

Extensions

Initial terms by John King and Mia Müßig added by Mia Muessig, Oct 05 2024

A374934 Maximum number of squares covered (i.e., attacked) by 4 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

16, 25, 36, 49, 62, 76, 92, 104, 120, 136, 152, 168, 184, 200, 216
Offset: 4

Views

Author

John King, Aug 08 2024

Keywords

Examples

			4 X 4:
  x Q x x
  x x x Q
  Q x x x
  x x Q x
5 X 5 there are several arrangements:
  x Q x x x
  x x x x x
  x x x x Q
  Q x x x x
  x x x Q x
6 X 6 and 7 X 7 (add a row and column) pattern as 4 queens knight-1,3 and 1,4 separation (not symmetric):
  . . . . . . .
  x x x x Q x .
  Q x x x x x .
  x x x x x x .
  x x x x x Q .
  x Q x x x x .
8 X 8: queens all knight-1,4 apart;
8 X 8 has 2 o/s;
9 X 9 has 5 o/s;
10 X 10 has 8 o/s;
  o x x x x x x x x o
  x o x x x x x x o x
  x x x Q x x x x x x
  x x x x x x x Q x x
  x x x x x x x x x x
  x x x x x x x x x x
  x x Q x x x x x x x
  x x x x x x Q x x x
  x o x x x x x x o x
  o x x x x x x x x o
beyond 10 X 10, the 4 queens separated as 1,2 knights begins to be the best layout; at 15 X 15, the pattern is clear.
  o x x o o x x x x o o x x o x
  x o x x o x x x x o x x o x x
  x x o x x x x x x x x o x x o
  o x x o x x x x x x o x x o o
  o o x x o x x x x o x x o o o
  x x x x x x Q x x x x x x x x
  x x x x x x x x Q x x x x x x
  x x x x x Q x x x x x x x x x
  x x x x x x x Q x x x x x x x
  o o x x o x x x x o x x o o o
  o x x o x x x x x x o x x o o
  x x o x x x x x x x x o x x o
  x o x x o x x x x o x x o x x
  o x x o o x x x x o o x x o x
  x x o o o x x x x o o o x x o

Crossrefs

Column 4 of A376732.
Cf. A017113, A047461, A374933, A375116, A374935, A374936, A374937, A374938.

Extensions

a(18) added using data from Mia Muessig by Andrew Howroyd, Oct 05 2024

A374936 Maximum number of squares covered (i.e., attacked) by 6 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

36, 49, 64, 81, 100, 121, 142, 165, 186, 209, 231, 255, 277
Offset: 6

Views

Author

John King, Aug 08 2024

Keywords

Examples

			Example for 12 X 12: There are 2 cells marked 'o' or uncovered thus a(12) = 12 * 12 - 2 = 142.
  x x x x x x x x x x x Q
  x x x x x x x x x x x x
  x x x x x x x x x x x x
  x x x Q x x x x x x x x
  x x x x x Q x x x x x x
  x x x x x x x Q x x x x
  x x x x Q x x x x x x x
  x x x x x x Q x x x x x
  o x x x x x x x x x x x
  x x x x x x x x x x x x
  x x x x x x x x x x x x
  x x x x x x x x o x x x
From _Christian Sievers_, Sep 08 2024: (Start)
Example for 14 X 14 with 186 attacked squares (unattacked ones marked with "+"):
  . . Q . . . . . . . . . . .
  . . . . . . . . . Q . . . .
  . . . . . . . . . . . . . +
  . + . . . . . . . . . . . .
  . . . Q . . . . . . . . . .
  . . . . . . . . . . . . . .
  . . . . . . . . . . . . . .
  . . . . . . . . . . . . Q .
  . + . . . . . . . . . . . .
  . . . . . . . . . . . . . +
  . . . . . . Q . . . . . . .
  . + . . + . . . . . . + . .
  . . . . . + . . . . + . . +
  Q . . . . . . . . . . . . .
(End)

Crossrefs

Column 6 of A376732.
Cf. A075324, A047461, A374933, A375116, A374933, A374934, A374935, A374937, A374938.

Extensions

a(14) corrected and a(15) confirmed by Christian Sievers, Sep 08 2024
a(16)-a(18) added using data from Mia Muessig by Andrew Howroyd, Oct 05 2024
Showing 1-3 of 3 results.

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