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.

User: John King

John King's wiki page.

John King has authored 13 sequences. Here are the ten most recent ones:

A376998 Minimum number of matchsticks required to make equilateral triangles of side lengths 1, 2, ..., n simultaneously.

Original entry on oeis.org

3, 7, 12, 17, 23
Offset: 1

Views

Author

John King, Oct 20 2024

Keywords

Comments

Upper bounds for a(6)-a(15) are 29, 35, 41, 48, 55, 62, 69, 77, 85, 93.
The triangles are allowed to be oriented upside-down. All triangles should be within the largest triangle.
This is the triangle equivalent of A294249.

Links

Crossrefs

Cf. A294249.

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

A374938 Maximum number of squares covered (i.e., attacked) by 8 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

64, 81, 100, 121, 144, 169, 196, 224, 251
Offset: 8

Views

Author

John King, Sep 09 2024

Keywords

Comments

Initial terms computed by Mia Muessig.

Crossrefs

Column 8 of A376732.

Formula

a(17) >= 281, a(18) >= 310, a(19) >= 338, a(20) >= 370.

A374937 Maximum number of squares covered (i.e., attacked) by 7 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

49, 64, 81, 100, 121, 144, 169, 194, 221, 242, 269, 294
Offset: 7

Views

Author

John King, Sep 09 2024

Keywords

Comments

Initial terms computed by Mia Muessig.

Crossrefs

Column 7 of A376732.

Formula

a(19) >= 321, a(20) >= 348.

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

A374935 Maximum number of squares covered (i.e., attacked) by 5 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

25, 36, 49, 64, 81, 100, 121, 134, 153, 172, 193, 212, 233, 252
Offset: 5

Views

Author

John King, Aug 08 2024

Keywords

Examples

			5 X 5; 6 X 6; 7 X 7; 8 X 8;  Center-square +4Queens separated as if 1,2 knights.
              at 11 X 11 and beyond this pattern seems to be 'best'.
  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 Q x x
  x x x Q x x x x
  x Q x x x x x x
  x x x x Q x x x
  x x x x x x x x
9 X 9; 10 X 10; 11 X 11; Center-square +4Queens separated as 2,4 knights.
  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 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 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 Q 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 x x x x x

Crossrefs

Column 5 of A376732.
Cf. A075324, A047461, A374933, A375116, A374933, A374934, A374936.

Extensions

Unverified a(19) removed by Andrew Howroyd, 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

A375116 Maximum number of squares covered (i.e., attacked) by 3 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

16, 25, 35, 45, 55, 66, 77, 88, 101, 112, 125, 136, 149, 160, 173, 184, 197, 208, 221, 232, 245, 256, 269, 280, 293, 304, 317, 328, 341, 352, 365, 376, 389, 400, 413, 424, 437, 448, 461, 472, 485, 496, 509, 520, 533, 544, 557, 568, 581, 592, 605, 616, 629, 640, 653, 664, 677
Offset: 4

Views

Author

John King, Jul 30 2024

Keywords

Comments

It is not possible to place 3 independent queens on a 1 X 1 or 2 X 2 or 3 X 3 board.
There is a related sequence of 'uncovered' squares i.e., n^2 - a(n).
There is another sequence denoting the potency of the new queen a(n) - A374933(n).

Examples

			4 X 4 complete coverage with 3 queens
  x x x x
  x Q x x
  x x x Q
  Q x x x
5 X 5 complete coverage with 3 queens
  Q x x x x
  x x x x x
  x x x Q x
  x x x x x
  x x Q x x
6 X 6 incomplete 1 o/s
  x x x x o x
  Q 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
6 X 6 coverage complete but NOT independent
  Q 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 q x x x
  x x x x x x
7 X 7 best leaves 4 o/s  (same layout as 6 X 6 with extra row and column)
There are alternative layouts - how many is not identified.
  x x x x o 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 Q x x x x
  x x x x x x o
  x x x o x x o

Crossrefs

Column 3 of A376732.
Cf. A047461 (for one queen), A374933 (for two queens), A374934, A374935, A374936.

Formula

a(n) = 12*n - 43 - (n mod 2) for n >= 10.

Extensions

a(6)-a(8) corrected by John King, Sep 17 2024
a(9) corrected using data from Mia Muessig by Andrew Howroyd, Oct 05 2024

A374933 Maximum number of squares covered (i.e., attacked) by 2 independent (i.e., non-attacking) queens on an n X n chessboard.

Original entry on oeis.org

9, 15, 23, 30, 37, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404, 412, 420
Offset: 3

Views

Author

John King, Jul 24 2024

Keywords

Comments

It is not possible to place two non-attacking queens on a 1 X 1 or 2 X 2 chessboard.

Links

Crossrefs

Cf. A017113, A047461 (case for one queen).

Formula

a(n) = 8*n - 20 for n >= 8.
G.f.: x^3*(9 - 3*x + 2*x^2 - x^3 + x^6)/(1 - x)^2. - Stefano Spezia, Jul 25 2024

A331208 The minimum perimeter for exactly n matchstick squares of size >= 1.

Original entry on oeis.org

0, 4, 6, 8, 10, 8, 10, 12, 10, 12, 14, 12, 14, 16, 12, 14, 16, 14, 16, 18, 14, 16, 18, 16, 18, 20, 16, 18, 18, 18, 16, 18, 20
Offset: 0

Views

Author

John King, Jan 12 2020

Keywords

Links

Crossrefs

Cf. A027709, A331207, A078633.

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

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