A218663 T(n,k) = Hilltop maps: number of n X k binary arrays indicating the locations of corresponding elements not exceeded by any king-move neighbor in a random 0..1 n X k array.
1, 3, 3, 5, 15, 5, 9, 57, 57, 9, 17, 225, 417, 225, 17, 31, 891, 3249, 3249, 891, 31, 57, 3519, 25533, 50625, 25533, 3519, 57, 105, 13905, 199489, 793881, 793881, 199489, 13905, 105, 193, 54945, 1560161, 12383361, 24879489, 12383361, 1560161, 54945
Offset: 1
Examples
Table starts ....1........3...........5...............9.................17 ....3.......15..........57.............225................891 ....5.......57.........417............3249..............25533 ....9......225........3249...........50625.............793881 ...17......891.......25533..........793881...........24879489 ...31.....3519......199489........12383361..........775176415 ...57....13905.....1560161.......193349025........24176619049 ..105....54945....12202673......3018953025.......754066017977 ..193...217107....95434773.....47135449449.....23517838102321 ..355...857871...746388537....735942652641....733484062428443 ..653..3389769..5837454753..11490533873361..22876204302519509 .1201.13394241.45654295713.179405691966081.713472099034206097 ... Some solutions for n=3 k=4 ..1..1..1..0....1..0..1..1....0..1..0..1....0..1..1..0....1..0..0..0 ..0..1..0..0....0..0..0..0....1..0..0..1....0..0..1..1....0..0..1..1 ..0..1..0..1....1..1..0..1....0..1..1..1....1..1..0..1....1..1..0..0
Links
- Stephan Mertens, Table of n, a(n) for n = 1..946 (first 240 terms from R. H. Hardin)
- Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
- Wikipedia, Dominating set
Crossrefs
Formula
Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2) +a(n-3)
k=2: a(n) = 3*a(n-1) +3*a(n-2) +3*a(n-3)
k=3: a(n) = 6*a(n-1) +11*a(n-2) +26*a(n-3) -5*a(n-4) -5*a(n-6)
k=4: a(n) = 12*a(n-1) +45*a(n-2) +180*a(n-3) -27*a(n-4) -81*a(n-6)
Columns k=1..z+1 for an underlying 0..z array: a(n) = sum(i=1..2z+1){(2^k-1)*a(n-i)} checked for z=1..3.
Comments