A303141
Number of minimum total dominating sets in the n X n black bishop graph.
Original entry on oeis.org
0, 1, 4, 1, 2, 61, 9, 104, 2586, 270, 5460, 184710, 12600, 458640, 20113380, 982800, 53373600, 3042295200, 109544400, 8497742400, 611967862800, 16922228400, 1762112167200, 157184659447200, 3432645216000, 463679536320000, 50284574580240000, 888196949640000, 150857156385936000, 19562163412303512000
Offset: 1
A303147
Number of total dominating sets in the n X n white bishop graph.
Original entry on oeis.org
1, 9, 143, 2699, 199095, 14042731, 3855890801, 1030417772377, 1084629728348393, 1128432868605656409, 4666227312488067563575, 19214059289771315688645819, 315759892137678954308707379391, 5181941387199963072391681357467099, 339917045534987610111076281503519527705
Offset: 2
A321673
Number of minimum total dominating sets in the n X n bishop graph.
Original entry on oeis.org
0, 1, 16, 1, 8, 3721, 108, 10816, 6568440, 72900, 31449600, 34117784100, 165564000, 210350649600, 402994899820800, 965895840000, 2867571383040000, 9255560083943040000, 12074509581120000, 72211625896757760000, 373919369489319044160000, 286361814021766560000, 3107808843778161008640000
Offset: 1
A303229
Number of minimal total dominating sets in the n X n white bishop graph.
Original entry on oeis.org
1, 4, 15, 72, 695, 5424, 66861, 721752
Offset: 2
A303230
Number of minimal total dominating sets in the n X n black bishop graph.
Original entry on oeis.org
0, 1, 5, 15, 113, 695, 6875, 66861, 933030
Offset: 1
A323501
Number of minimum dominating sets in the n X n white bishop graph.
Original entry on oeis.org
2, 6, 5, 2, 22, 356, 108, 24, 672, 25056, 4680, 720, 38160, 2531520, 342720, 40320, 3467520, 358444800, 38102400, 3628800, 460857600, 68388364800, 5987520000, 479001600, 84304281600, 16979648716800, 1264085222400, 87178291200, 20312541849600
Offset: 2
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Table[Piecewise[{{(n/2)! (n + 1)/2, Mod[n, 4] == 0}, {((n - 1)/2)!, Mod[n, 4] == 1}, {(n/2 - 1)! (n^2 + n + 2)/4, Mod[n, 4] == 2}, {((n - 3)/2)! (n + 1) (n^3 + n^2 - 6 n + 6)/16, Mod[n, 4] == 3}}], {n, 2, 20}] (* Eric W. Weisstein, Feb 27 2025 *)
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\\ See A289170 for DomSetCount, Bishop.
a(n)={Vec(DomSetCount(Bishop(n, 1), x + O(x^((n+3)\2))))[1]} \\ Andrew Howroyd, Sep 08 2019
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a(n)=(n\4*2)!*if(n%4<2, if(n%2==0, (n + 1)/2, 1), if(n%2==0, (n^2 + n + 2)/4, (n + 1)*(n^3 + n^2 - 6*n + 6)/16)); \\ Andrew Howroyd, Sep 09 2019
Offset corrected and terms a(11) and beyond from
Andrew Howroyd, Sep 08 2019
A381727
Number of minimum connected dominating sets in the n X n white bishop graph.
Original entry on oeis.org
2, 4, 1, 4, 13, 64, 513, 4480, 41197, 444416, 5597201, 77253632, 1153902701, 18870222848, 336018968449, 6428081455104, 131386321421901, 2865273888571392, 66426533670738769, 1629643279560867840, 42175861619149917325, 1148845693539400548352, 32856688248674995989889
Offset: 2
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Join[{2, 4}, Table[Sum[(2 k)^(n - 2 k - 2) (n - 2 k - 1)^(2 k - 1), {k, Floor[n/2] - 1}], {n, 4, 20}]] (* Eric W. Weisstein, Mar 22 2025 *)
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\\ B(n, k) is A072590.
B(n,k) = n^(k-1) * k^(n-1)
a(n) = if(n <= 3, 2*n-2, sum(k=1, n\2-1, B(n-1-2*k, 2*k))) \\ Andrew Howroyd, Mar 20 2025
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