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A101201 Maximal number of kings in the toroidal king's graph on an n X n board such that each king is attacking no more than four other kings. 0
0, 2, 5, 9, 15, 21, 28, 37, 47, 60, 71, 84 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
All the numbers listed so far were calculated using LpSolveIDE. It can be shown that the sequence of densities a(n)/n^2 has a limit as n goes to infinity, which is the supremum of all the elements in the sequence. With the help of the computer, it was shown that a(n) is not more than 0.608956n^2.
LINKS
E. J. Ionascu, D. Pritkin and S. E. Wright, k-Dependence and Domination in Kings Graphs, Amer. Math. Monthly, vol. 115, no. 9, (2008), pp. 820-836.
FORMULA
a(n) is approximately 3n^2/5 (conjecture).
EXAMPLE
a(2)=2 because one can check that any arrangement of two kings will satisfy the requirement but any arrangement of three kings will not.
PROG
(LpSolveIDE) /* Objective function n=10*/
max: x11 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + x19 + x110
+ x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28 + x29 + x210
+ x31 + x32 + x33 + x34 + x35 + x36 + x37 + x38 + x39 + x310
+ x41 + x42 + x43 + x44 + x45 + x46 + x47 + x48 + x49 + x410
+ x51 + x52 + x53 + x54 + x55 + x56 + x57 + x58 + x59 + x510
+ x61 + x62 + x63 + x64 + x65 + x66 + x67 + x68 + x69 + x610
+ x71 + x72 + x73 + x74 + x75 + x76 + x77 + x78 + x79 + x710
+ x81 + x82 + x83 + x84 + x85 + x86 + x87 + x88 + x89 + x810
+ x91 + x92 + x93 + x94 + x95 + x96 + x97 + x98 + x99 + x910
+ x101 + x102 + x103 + x104 + x105 + x106 + x107 + x108 + x109 + x1010;
/* Variable bounds */
2x11 + x12 + x22 + x21 < 3;
2x110 + x19 + x29 + x210 < 3;
2x101 + x91 + x92 + x102 < 3;
2x1010 + x109 + x99 + x910 < 3;
3x12 + x11 + x21 + x22 + x23 + x13 < 5;
3x13 + x12 + x22 + x23 + x24 + x14 < 5;
3x14 + x13 + x23 + x24 + x25 + x15 < 5;
3x15 + x14 + x24 + x25 + x26 + x16 < 5;
3x16 + x15 + x25 + x26 + x27 + x17 < 5;
3x17 + x16 + x26 + x27 + x28 + x18 < 5;
3x18 + x17 + x27 + x28 + x29 + x19 < 5;
3x19 + x18 + x28 + x29 + x210 + x110 < 5;
3x21 + x11 + x12 + x22 + x32 + x31 < 5;
3x31 + x21 + x22 + x32 + x42 + x41 < 5;
3x41 + x31 + x32 + x42 + x52 + x51 < 5;
3x51 + x41 + x42 + x52 + x62 + x61 < 5;
3x61 + x51 + x52 + x62 + x72 + x71 < 5;
3x71 + x61 + x62 + x72 + x82 + x81 < 5;
3x81 + x71 + x72 + x82 + x92 + x91 < 5;
3x91 + x81 + x82 + x92 + x102 + x101 < 5;
3x210 + x110 + x19 + x29 + x39 + x310 < 5;
3x310 + x210 + x29 + x39 + x49 + x410 < 5;
3x410 + x310 + x39 + x49 + x59 + x510 < 5;
3x510 + x410 + x49 + x59 + x69 + x610 < 5;
3x610 + x510 + x59 + x69 + x79 + x710 < 5;
3x710 + x610 + x69 + x79 + x89 + x810 < 5;
3x810 + x710 + x79 + x89 + x99 + x910 < 5;
3x910 + x810 + x89 + x99 + x109 + x1010 < 5;
3x102 + x101 + x91 + x92 + x93 + x103 < 5;
3x103 + x102 + x92 + x93 + x94 + x104 < 5;
3x104 + x103 + x93 + x94 + x95 + x105 < 5;
3x105 + x104 + x94 + x95 + x96 + x106 < 5;
3x106 + x105 + x95 + x96 + x97 + x107 < 5;
3x107 + x106 + x96 + x97 + x98 + x108 < 5;
3x108 + x107 + x97 + x98 + x99 + x109 < 5;
3x109 + x108 + x98 + x99 + x910 + x1010 < 5;
4x22 + x11 + x12 + x13 + x21 + x23 + x31 + x32 + x33 < 8;
4x23 + x12 + x13 + x14 + x22 + x24 + x32 + x33 + x34 < 8;
4x24 + x13 + x14 + x15 + x23 + x25 + x33 + x34 + x35 < 8;
4x25 + x14 + x15 + x16 + x24 + x26 + x34 + x35 + x36 < 8;
4x26 + x15 + x16 + x17 + x25 + x27 + x35 + x36 + x37 < 8;
4x27 + x16 + x17 + x18 + x26 + x28 + x36 + x37 + x38 < 8;
4x28 + x17 + x18 + x19 + x27 + x29 + x37 + x38 + x39 < 8;
4x29 + x18 + x19 + x110 + x28 + x210 + x38 + x39 + x310 < 8;
4x32 + x21 + x22 + x23 + x31 + x33 + x41 + x42 + x43 < 8;
4x33 + x22 + x23 + x24 + x32 + x34 + x42 + x43 + x44 < 8;
4x34 + x23 + x24 + x25 + x33 + x35 + x43 + x44 + x45 < 8;
4x35 + x24 + x25 + x26 + x34 + x36 + x44 + x45 + x46 < 8;
4x36 + x25 + x26 + x27 + x35 + x37 + x45 + x46 + x47 < 8;
4x37 + x26 + x27 + x28 + x36 + x38 + x46 + x47 + x48 < 8;
4x38 + x27 + x28 + x29 + x37 + x39 + x47 + x48 + x49 < 8;
4x39 + x28 + x29 + x210 + x38 + x310 + x48 + x49 + x410 < 8;
4x42 + x31 + x32 + x33 + x41 + x43 + x51 + x52 + x53 < 8;
4x43 + x32 + x33 + x34 + x42 + x44 + x52 + x53 + x54 < 8;
4x44 + x33 + x34 + x35 + x43 + x45 + x53 + x54 + x55 < 8;
4x45 + x34 + x35 + x36 + x44 + x46 + x54 + x55 + x56 < 8;
4x46 + x35 + x36 + x37 + x45 + x47 + x55 + x56 + x57 < 8;
4x47 + x36 + x37 + x38 + x46 + x48 + x56 + x57 + x58 < 8;
4x48 + x37 + x38 + x39 + x47 + x49 + x57 + x58 + x59 < 8;
4x49 + x38 + x39 + x310 + x48 + x410 + x58 + x59 + x510 < 8;
4x52 + x41 + x42 + x43 + x51 + x53 + x61 + x62 + x63 < 8;
4x53 + x42 + x43 + x44 + x52 + x54 + x62 + x63 + x64 < 8;
4x54 + x43 + x44 + x45 + x53 + x55 + x63 + x64 + x65 < 8;
4x55 + x44 + x45 + x46 + x54 + x56 + x64 + x65 + x66 < 8;
4x56 + x45 + x46 + x47 + x55 + x57 + x65 + x66 + x67 < 8;
4x57 + x46 + x47 + x48 + x56 + x58 + x66 + x67 + x68 < 8;
4x58 + x47 + x48 + x49 + x57 + x59 + x67 + x68 + x69 < 8;
4x59 + x48 + x49 + x410 + x58 + x510 + x68 + x69 + x610 < 8;
4x62 + x51 + x52 + x53 + x61 + x63 + x71 + x72 + x73 < 8;
4x63 + x52 + x53 + x54 + x62 + x64 + x72 + x73 + x74 < 8;
4x64 + x53 + x54 + x55 + x63 + x65 + x73 + x74 + x75 < 8;
4x65 + x54 + x55 + x56 + x64 + x66 + x74 + x75 + x76 < 8;
4x66 + x55 + x56 + x57 + x65 + x67 + x75 + x76 + x77 < 8;
4x67 + x56 + x57 + x58 + x66 + x68 + x76 + x77 + x78 < 8;
4x68 + x57 + x58 + x59 + x67 + x69 + x77 + x78 + x79 < 8;
4x69 + x58 + x59 + x510 + x68 + x610 + x78 + x79 + x710 < 8;
4x72 + x61 + x62 + x63 + x71 + x73 + x81 + x82 + x83 < 8;
4x73 + x62 + x63 + x64 + x72 + x74 + x82 + x83 + x84 < 8;
4x74 + x63 + x64 + x65 + x73 + x75 + x83 + x84 + x85 < 8;
4x75 + x64 + x65 + x66 + x74 + x76 + x84 + x85 + x86 < 8;
4x76 + x65 + x66 + x67 + x75 + x77 + x85 + x86 + x87 < 8;
4x77 + x66 + x67 + x68 + x76 + x78 + x86 + x87 + x88 < 8;
4x78 + x67 + x68 + x69 + x77 + x79 + x87 + x88 + x89 < 8;
4x79 + x68 + x69 + x610 + x78 + x710 + x88 + x89 + x810 < 8;
4x82 + x71 + x72 + x73 + x81 + x83 + x91 + x92 + x93 < 8;
4x83 + x72 + x73 + x74 + x82 + x84 + x92 + x93 + x94 < 8;
4x84 + x73 + x74 + x75 + x83 + x85 + x93 + x94 + x95 < 8;
4x85 + x74 + x75 + x76 + x84 + x86 + x94 + x95 + x96 < 8;
4x86 + x75 + x76 + x77 + x85 + x87 + x95 + x96 + x97 < 8;
4x87 + x76 + x77 + x78 + x86 + x88 + x96 + x97 + x98 < 8;
4x88 + x77 + x78 + x79 + x87 + x89 + x97 + x98 + x99 < 8;
4x89 + x78 + x79 + x710 + x88 + x810 + x98 + x99 + x910 < 8;
4x92 + x81 + x82 + x83 + x91 + x93 + x101 + x102 + x103 < 8;
4x93 + x82 + x83 + x84 + x92 + x94 + x102 + x103 + x104 < 8;
4x94 + x83 + x84 + x85 + x93 + x95 + x103 + x104 + x105 < 8;
4x95 + x84 + x85 + x86 + x94 + x96 + x104 + x105 + x106 < 8;
4x96 + x85 + x86 + x87 + x95 + x97 + x105 + x106 + x107 < 8;
4x97 + x86 + x87 + x88 + x96 + x98 + x106 + x107 + x108 < 8;
4x98 + x87 + x88 + x89 + x97 + x99 + x107 + x108 + x109 < 8;
4x99 + x88 + x89 + x810 + x98 + x910 + x108 + x109 + x1010 < 8;
x11 < 1; x12 < 1; x13 < 1; x14 < 1; x15 < 1; x16 < 1; x17 < 1; x18 < 1; x19 < 1; x110 < 1;
x21 < 1; x22 < 1; x23 < 1; x24 < 1; x25 < 1; x26 < 1; x27 < 1; x28 < 1; x29 < 1; x210 < 1;
x31 < 1; x32 < 1; x33 < 1; x34 < 1; x35 < 1; x36 < 1; x37 < 1; x38 < 1; x39 < 1; x310 < 1;
x41 < 1; x42 < 1; x43 < 1; x44 < 1; x45 < 1; x46 < 1; x47 < 1; x48 < 1; x49 < 1; x410 < 1;
x51 < 1; x52 < 1; x53 < 1; x54 < 1; x55 < 1; x56 < 1; x57 < 1; x58 < 1; x59 < 1; x510 < 1;
x61 < 1; x62 < 1; x63 < 1; x64 < 1; x65 < 1; x66 < 1; x67 < 1; x68 < 1; x69 < 1; x610 < 1;
x71 < 1; x72 < 1; x73 < 1; x74 < 1; x75 < 1; x76 < 1; x77 < 1; x78 < 1; x79 < 1; x710 < 1;
x81 < 1; x82 < 1; x83 < 1; x84 < 1; x85 < 1; x86 < 1; x87 < 1; x88 < 1; x89 < 1; x810 < 1;
x91 < 1; x92 < 1; x93 < 1; x94 < 1; x95 < 1; x96 < 1; x97 < 1; x98 < 1; x99 < 1; x910 < 1;
x101 < 1; x102 < 1; x103 < 1; x104 < 1; x105 < 1; x106 < 1; x107 < 1; x108 < 1; x109 < 1; x1010 < 1;
int x11, x12, x13, x14, x15, x16, x17, x18, x19, x110,
x21, x22, x23, x24, x25, x26, x27, x28, x29, x210,
x31, x32, x33, x34, x35, x36, x37, x38, x39, x310,
x41, x42, x43, x44, x45, x46, x47, x48, x49, x410,
x51, x52, x53, x54, x55, x56, x57, x58, x59, x510,
x61, x62, x63, x64, x65, x66, x67, x68, x69, x610,
x71, x72, x73, x74, x75, x76, x77, x78, x79, x710,
x81, x82, x83, x84, x85, x86, x87, x88, x89, x810,
x91, x92, x93, x94, x95, x96, x97, x98, x99, x910,
x101, x102, x103, x104, x105, x106, x107, x108, x109, x1010;
CROSSREFS
Sequence in context: A051892 A006599 A013933 * A184535 A033096 A195014
KEYWORD
nonn,more
AUTHOR
Eugen J. Ionascu, Aug 12 2008
STATUS
approved

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Last modified March 29 10:22 EDT 2024. Contains 371268 sequences. (Running on oeis4.)