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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

Table of n, a(n) for n=1..12.

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

Cf. A103139, A189889.

Sequence in context: A051892 A006599 A013933 * A184535 A033096 A195014

Adjacent sequences:  A101198 A101199 A101200 * A101202 A101203 A101204

KEYWORD

nonn,more

AUTHOR

Eugen J. Ionascu, Aug 12 2008

STATUS

approved

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Last modified November 19 10:44 EST 2017. Contains 294936 sequences.