login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A274947 Irregular triangle read by rows: T(n,k) (n>=0, 0 <= k <= n^2) = least number of squares attacked by k queens on an n X n board. 4
0, 0, 1, 0, 4, 4, 4, 4, 0, 7, 8, 9, 9, 9, 9, 9, 9, 9, 0, 10, 13, 14, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 0, 13, 18, 20, 21, 22, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 0, 16, 23, 27, 28, 30, 31, 32, 32, 33, 34, 34, 34, 34, 35, 35, 35, 35, 35, 35, 35, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Place k queens on an n X n board so that the total number of squares attacked/occupied by the queens is minimized.
If enough terms were known, would provide an upper bound for A250000. For if A250000(n) = Q then T(n,Q) <= n^2 - Q, or equivalently A274948(n,Q) >= Q.
Values n^2 - T(n,n) are given in A001366.
Let X(n) be the smallest number so that no matter how you place X queens, they attack every square. That is, X is the minimal number such that T(n,k) = n^2 for all k >= X. Then X = n^2 - T(n,1) + 1 = A274948(n,1) + 1 = n^2 - 3*n + 3. More generally, T(n,k') <= n^2-k if and only if k' <= n^2-T(n,k). For example, we may place 2 queens on two squares of a 4 X 4 board and leave 4^2-T(4,2)=3 squares not attacked, so we may place 3 queens on these 3 squares instead and leave those two squares not attacked, ergo, T(4,3)=16-2. - Andrey Zabolotskiy, Jul 29 2016
LINKS
Bernard Lemaire and Pavel Vitushinkiy, Placing n non dominating queens on the n X n chessboard. Part I, French Federation of Mathematical Games.
Bernard Lemaire and Pavel Vitushinkiy, Placing n non dominating queens on the n X n chessboard. Part II, French Federation of Mathematical Games.
FORMULA
T(n,1) = 3*n-2 for n >= 1.
EXAMPLE
The triangle begins:
0
0, 1,
0, 4, 4, 4, 4,
0, 7, 8, 9, 9, 9, 9, 9, 9, 9,
0, 10, 13, 14, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,
0, 13, 18, 20, 21, 22, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25,
0, 16, 23, 27, 28, 30, 31, 32, 32, 33, 34, 34, 34, 34, 35, 35, 35, 35, 35, 35, 35, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36,
0, 19, 28, 33, 33, 38, 39, 42, 43, 43, 43, 44, 45, 45, 45, 45, 45, 47, 47, 47, 47, 47, 48, 48, 48, 48, 48, 48, 48, 48, 48, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49,
0, 22, 33, 39, 40, 47, 49, 51, 53, 54, 55, 56, 57, 57, 58, 58, 59, 59, 60, 60, 60, 60, 60, 60, 60, 61, 62, 62, 62, 62, 62, 62, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,
0, 25, 38, 45, 45, 54, 57, 61, 62, 63, 67, 68, 69, 70, 71, 72, 72, 72, 72, 73, 74, 75, 75, 75, 75, 76, 76, 76, 77, 77, 77, 77, 77, 77, 77, 77, 77, 79, 79, 79, 79, 79, 79, 79, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81,
0, 28, 43, 51, 52, 63, 67, 70, 74, 76, 78, 81, 82, 84, 85, 86, 87, 88, 88, 89, 90, 90, 90, 91, 91, 92, 92, 93, 93, 93, 93, 94, 94, 94, 95, 95, 95, 95, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 97, 98, 98, 98, 98, 98, 98, 98, 98, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100,
...
(Rows 6 through 10 from Rob Pratt, Aug 02 2016)
The entry T(4,3) = 14 is achieved by
OXOX
OOOX
AOOO
OOAO
since the two squares marked A are not attacked by the three queens at X.
CROSSREFS
Cf. A075458 (minimal number of queens needed to attack all the squares of an n X n board).
Row 8 subtracted from 64 is A342151.
Sequence in context: A174987 A214926 A268368 * A331961 A279406 A171408
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jul 27 2016
EXTENSIONS
Corrections and more terms from Andrey Zabolotskiy, Jul 29 2016
More terms via integer linear programming from Rob Pratt, Aug 02 2016
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)