A217632 -id:A217632 - cp's OEIS frontend

A217637 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nXk array.

2, 2, 2, 4, 6, 4, 6, 16, 16, 6, 10, 38, 66, 38, 10, 16, 98, 244, 244, 98, 16, 26, 244, 968, 1418, 968, 244, 26, 42, 614, 3726, 8706, 8706, 3726, 614, 42, 68, 1542, 14520, 52120, 83074, 52120, 14520, 1542, 68, 110, 3872, 56352, 315378, 773348, 773348, 315378

Offset: 1

R. H. Hardin, Oct 09 2012

Number of maximal independent sets in the graph P_2 X P_n X P_k. - Andrew Howroyd, Jun 10 2017

			Table starts
...2.....2........4..........6...........10..............16................26
...2.....6.......16.........38...........98.............244...............614
...4....16.......66........244..........968............3726.............14520
...6....38......244.......1418.........8706...........52120............315378
..10....98......968.......8706........83074..........773348...........7272142
..16...244.....3726......52120.......773348........11181454.........163361868
..26...614....14520.....315378......7272142.......163361868........3709621842
..42..1542....56352....1900838.....68138974......2378097084.......83923710538
..68..3872...218978...11472148....639248556.....34661572702.....1901055652804
.110..9726...850620...69210290...5994907930....505010822224....43046530809006
.178.24426..3304624..417586442..56226693158...7358779655656...974841850791586
.288.61348.12837742.2519466108.527340415924.107224919634686.22075731493018104
...
Some solutions for n=3 k=4
..1..0..0..1....0..0..0..1....1..0..1..1....1..1..0..0....1..0..0..0
..0..0..0..0....0..0..1..1....0..0..0..1....0..0..1..0....1..1..0..0
..1..0..0..0....0..0..0..1....1..0..1..1....0..0..0..1....1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..220
MacKenzie Carr, Christina M. Mynhardt, Ortrud R. Oellermann, Enumerating the Digitally Convex Sets of Powers of Cycles and Cartesian Products of Paths and Complete Graphs, arXiv:2008.02781 [math.CO], 2020.
R. Euler, P. Oleksik, Z. Skupien, Counting Maximal Distance-Independent Sets in Grid Graphs, Discussiones Mathematicae Graph Theory. Volume 33, Issue 3, Pages 531-557, ISSN (Print) 2083-5892, July 2013; see also.

Columns 1-3 are A006355(n+1), A217631, A217632.

Cf. A197054.

A217631 Number of nX2 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX2 array.

Original entry on oeis.org

0, 2, 6, 16, 38, 98, 244, 614, 1542, 3872, 9726, 24426, 61348, 154078, 386974, 971904, 2440982, 6130642, 15397396, 38671286, 97124758, 243933408, 612650254, 1538699994, 3864517572, 9705918062, 24376870766, 61223660096, 153766108518

Offset: 0

R. H. Hardin Oct 09 2012

nonn

Also, number of maximal independent sets in the 3-dimensional (2, 2, n) grid graph. [Euler et al.] - N. J. A. Sloane, Nov 21 2013

Column 2 of A217637.

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

R. H. Hardin, Table of n, a(n) for n = 0..210
R. Euler, P. Oleksik, Z. Skupien, Counting Maximal Distance-Independent Sets in Grid Graphs, Discussiones Mathematicae Graph Theory. Volume 33, Issue 3, Pages 531-557, ISSN (Print) 2083-5892, July 2013; http://www.degruyter.com/view/j/dmgt.2013.33.issue-3/dmgt.1707/dmgt.1707.xml

Cf. A217632, A217637.

G.f. = (2*x+4*x^2+4*x^3)/(1-x-3*x^2-2*x^3). [Euler et al.] - N. J. A. Sloane, Nov 21 2013

Empirical: a(n) = a(n-1) + 3*a(n-2) + 2*a(n-3). (Follows from g.f. - N. J. A. Sloane, Nov 21 2013)

A231884 Number of maximal 2-independent sets in the 3-dimensional (2, 2, n) grid graph.

Original entry on oeis.org

0, 4, 4, 20, 32, 80, 180, 408, 940, 2072, 4824, 10792, 24660, 55748, 126760, 287584, 652280, 1481184, 3359900, 7627296, 17305472, 39277688, 89131928, 202276640, 459045772, 1041743020, 2364140452, 5365103100, 12175556108, 27630957644, 62705400664, 142302685268

Offset: 0

N. J. A. Sloane, Nov 17 2013

Andrew Howroyd, Table of n, a(n) for n = 0..200
R. Euler, P. Oleksik, Z. Skupien, Counting Maximal Distance-Independent Sets in Grid Graphs, Discussiones Mathematicae Graph Theory. Volume 33, Issue 3, Pages 531-557, ISSN (Print) 2083-5892, July 2013; see also.
Index entries for linear recurrences with constant coefficients, signature (0,3,4,4,0,-9,-3).

Cf. A197054, A231886, A231887, A231888, A217631, A217632.

Join[{0}, LinearRecurrence[{0, 3, 4, 4, 0, -9, -3}, {4, 4, 20, 32, 80, 180, 408}, 31]] (* Jean-François Alcover, Nov 01 2017 *)

concat(0, Vec(4*x*(1 + x)*(1 + 2*x^2 - x^3 - 2*x^4 - x^5) / (1 - 3*x^2 - 4*x^3 - 4*x^4 + 9*x^6 + 3*x^7) + O(x^40))) \\ Colin Barker, Oct 04 2017

Euler et al. give an explicit g.f. and recurrence.
From Colin Barker, Oct 04 2017: (Start)
G.f.: 4*x*(1 + x)*(1 + 2*x^2 - x^3 - 2*x^4 - x^5) / (1 - 3*x^2 - 4*x^3 - 4*x^4 + 9*x^6 + 3*x^7).
a(n) = 3*a(n-2) + 4*a(n-3) + 4*a(n-4) - 9*a(n-6) - 3*a(n-7) for n>7.
(End)

Terms a(11) and beyond from Andrew Howroyd, Jun 10 2017

A231887 Number of maximal 2-independent sets in the 3-dimensional (3, 3, n) grid graph.

Original entry on oeis.org

0, 11, 46, 182, 1026, 4836, 23922, 118674, 584516, 2889306, 14266546, 70455052, 347980122, 1718525298, 8487343508, 41916544250, 207013446378, 1022380190332, 5049238367202, 24936725579450, 123155267567884, 608228181611074, 3003862808227186, 14835208208589988

Offset: 0

N. J. A. Sloane, Nov 17 2013

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100
R. Euler, P. Oleksik, Z. Skupien, Counting Maximal Distance-Independent Sets in Grid Graphs, Discussiones Mathematicae Graph Theory. Volume 33, Issue 3, Pages 531-557, ISSN (Print) 2083-5892, July 2013; see also.

Cf. A197054, A231884, A231886, A231888, A217631, A217632.

Terms a(13) and beyond from Andrew Howroyd, Jun 10 2017

A231888 Number of maximal 3-independent sets in the 3-dimensional (3, 3, n) grid graph.

Original entry on oeis.org

0, 7, 26, 57, 190, 646, 1914, 5960, 18824, 58248, 181196, 565328, 1759720, 5477376, 17062956, 53131824, 165443096, 515219344, 1604410136, 4996129756, 15558217340, 48448930636, 150871392396, 469819443204, 1463035474136, 4555944690588, 14187380873768

Offset: 0

N. J. A. Sloane, Nov 17 2013

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100
R. Euler, P. Oleksik, Z. Skupien, Counting Maximal Distance-Independent Sets in Grid Graphs, Discussiones Mathematicae Graph Theory. Volume 33, Issue 3, Pages 531-557, ISSN (Print) 2083-5892, July 2013; see also.

Cf. A197054, A231884, A231887, A231888, A217631, A217632.

Terms a(13) and beyond from Andrew Howroyd, Jun 10 2017

cp's OEIS Frontend

A217637 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nXk array.

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A217631 Number of nX2 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX2 array.

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A231884 Number of maximal 2-independent sets in the 3-dimensional (2, 2, n) grid graph.

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A231887 Number of maximal 2-independent sets in the 3-dimensional (3, 3, n) grid graph.

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A231888 Number of maximal 3-independent sets in the 3-dimensional (3, 3, n) grid graph.

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