A218426 -id:A218426 - cp's OEIS frontend

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

R. H. Hardin, Nov 04 2012

From Andrew Howroyd, May 10 2017: (Start)
Number of n X k binary matrices with every 1 adjacent to some 0 horizontally, vertically, diagonally or antidiagonally.
Number of dominating sets in the n X k king graph. (End)

			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

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

Columns 1-7 are A000213(n+1), A218657, A218658, A218659, A218660, A218661, A218662.
Diagonal is A133791.
Cf. A218354, A221446, A219078, A218426.

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.

A218420 Hilltop maps: number of n X 2 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 n X 2 array.

Original entry on oeis.org

3, 13, 49, 191, 737, 2849, 11011, 42557, 164481, 635711, 2456993, 9496161, 36702211, 141852301, 548252401, 2118969471, 8189716289, 31652864193, 122336815619, 472825977597, 1827449929601, 7063007117695, 27298186799297, 105506194473793

Offset: 1

R. H. Hardin, Oct 28 2012

nonn

Column 2 of A218426.

			Some solutions for n=3:
..1..1....0..1....1..1....0..0....0..1....0..0....1..0....1..1....0..1....1..0
..0..0....1..1....0..0....1..1....0..1....1..0....0..0....1..0....1..0....1..0
..0..1....0..1....1..1....1..1....1..1....1..1....1..0....1..0....1..1....1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A218426.

Empirical: a(n) = 3*a(n-1) + 3*a(n-2) + a(n-3) + a(n-4) + a(n-5).

Empirical g.f.: x*(3 + 4*x + x^2 + 2*x^3 + x^4) / (1 - 3*x - 3*x^2 - x^3 - x^4 - x^5). - Colin Barker, Mar 10 2018

A218419 Hilltop maps: number of n X n binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 n X n array.

Original entry on oeis.org

1, 13, 357, 41515, 19003811, 34328798853, 245489633250861, 6942897327251230265, 776657742366634012014603, 343639983448734808976866431015, 601395784904377113605427806277106571

Offset: 1

R. H. Hardin Oct 28 2012

nonn

Diagonal of A218426

			Some solutions for n=3
..1..1..1....1..1..0....1..1..1....0..0..0....0..1..1....1..1..0....0..1..0
..0..1..1....0..1..0....1..1..1....1..1..0....0..1..1....1..0..1....1..0..0
..1..0..1....0..1..1....0..1..1....1..1..0....0..0..1....0..0..0....1..1..1

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

A218421 Hilltop maps: number of nX3 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX3 array.

Original entry on oeis.org

5, 49, 357, 2725, 20793, 158429, 1207525, 9203377, 70144889, 534620525, 4074695269, 31055936969, 236697756661, 1804029548133, 13749697742817, 104795505274517, 798715588594961, 6087537722095689, 46397135660186377

Offset: 1

R. H. Hardin Oct 28 2012

nonn

Column 3 of A218426

			Some solutions for n=3
..0..1..0....1..0..0....0..1..1....1..1..0....1..0..0....0..1..0....0..1..1
..1..1..1....1..1..0....0..0..0....0..1..1....0..0..1....1..0..1....0..0..0
..1..1..1....1..0..1....0..1..1....0..1..1....1..0..0....0..0..0....1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 5*a(n-1) +16*a(n-2) +28*a(n-3) +17*a(n-4) +7*a(n-5) -3*a(n-6) +3*a(n-7) -8*a(n-8) -a(n-10) +a(n-11)

A218422 Hilltop maps: number of nX4 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX4 array.

Original entry on oeis.org

9, 191, 2725, 41515, 629659, 9539915, 144597047, 2191517757, 33214850067, 503408006597, 7629705089721, 115636627121941, 1752601092641329, 26562609636801375, 402585753260380041, 6101632743611966643

Offset: 1

R. H. Hardin Oct 28 2012

nonn

Column 4 of A218426

			Some solutions for n=3
..1..1..0..0....1..1..0..1....0..1..0..1....1..1..0..0....0..1..0..1
..1..0..1..0....0..1..1..1....1..1..1..1....1..0..1..1....1..0..0..1
..1..1..0..1....0..1..0..0....0..0..1..1....1..0..1..1....0..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..146

Empirical: a(n) = 9*a(n-1) +72*a(n-2) +284*a(n-3) +526*a(n-4) +890*a(n-5) +902*a(n-6) +1382*a(n-7) -553*a(n-8) +2141*a(n-9) -1904*a(n-10) +416*a(n-11) -53*a(n-12) -263*a(n-13) -38*a(n-14) +38*a(n-15) -6*a(n-16) -18*a(n-17) +16*a(n-18) -a(n-20) +a(n-21)

A218423 Hilltop maps: number of nX5 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX5 array.

Original entry on oeis.org

17, 737, 20793, 629659, 19003811, 572765605, 17270172871, 520703946625, 15699497444771, 473348740584743, 14271727157207365, 430300515162577777, 12973799949967137925, 391167286570698841921, 11793911323529489973393

Offset: 1

R. H. Hardin Oct 28 2012

nonn

Column 5 of A218426

			Some solutions for n=3
..0..0..0..0..1....0..1..0..1..1....1..0..1..0..0....1..1..0..1..0
..1..1..0..0..0....0..0..1..1..1....0..0..0..1..1....0..1..0..0..1
..1..1..1..0..1....0..1..1..0..1....1..0..0..1..1....0..1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..64

A218424 Hilltop maps: number of nX6 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX6 array.

Original entry on oeis.org

31, 2849, 158429, 9539915, 572765605, 34328798853, 2058493362799, 123428176573409, 7400808508179025, 443756537313437079, 26607874601757021869, 1595422183998461122637, 95662355032495391763215

Offset: 1

R. H. Hardin Oct 28 2012

nonn

Column 6 of A218426

			Some solutions for n=3
..1..1..1..1..1..0....1..1..1..0..0..0....0..1..0..0..1..0....1..1..0..0..1..0
..0..1..1..0..1..1....0..1..1..1..1..0....0..0..1..0..1..0....0..1..0..1..0..1
..0..1..1..0..1..0....1..1..0..1..0..1....0..1..0..0..1..0....0..1..0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..44

A218425 Hilltop maps: number of nX7 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX7 array.

Original entry on oeis.org

57, 11011, 1207525, 144597047, 17270172871, 2058493362799, 245489633250861, 29274579482710357, 3490981608801331021, 416298960473587173025, 49643559085814354624329, 5919983699927241114279393

Offset: 1

R. H. Hardin Oct 28 2012

nonn

Column 7 of A218426

			Some solutions for n=3
..0..1..0..1..0..1..1....0..1..1..1..1..0..0....0..1..0..1..0..1..1
..0..0..0..1..1..1..0....0..1..1..1..1..1..0....0..1..0..0..1..1..1
..0..1..1..1..1..0..1....0..0..0..1..1..0..1....0..0..1..0..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..83

cp's OEIS Frontend

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.

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A218420 Hilltop maps: number of n X 2 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 n X 2 array.

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A218419 Hilltop maps: number of n X n binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 n X n array.

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A218421 Hilltop maps: number of nX3 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX3 array.

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A218422 Hilltop maps: number of nX4 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX4 array.

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A218423 Hilltop maps: number of nX5 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX5 array.

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A218424 Hilltop maps: number of nX6 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX6 array.

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A218425 Hilltop maps: number of nX7 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX7 array.

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