A197049 -id:A197049 - cp's OEIS frontend

A197054 T(n,k)=Number of nXk 0..4 arrays with each element equal to the number of its horizontal and vertical zero neighbors.

1, 2, 2, 2, 2, 2, 3, 4, 4, 3, 4, 6, 10, 6, 4, 5, 10, 18, 18, 10, 5, 7, 16, 38, 42, 38, 16, 7, 9, 26, 78, 108, 108, 78, 26, 9, 12, 42, 156, 274, 358, 274, 156, 42, 12, 16, 68, 320, 692, 1132, 1132, 692, 320, 68, 16, 21, 110, 654, 1754, 3580, 4468, 3580, 1754, 654, 110, 21, 28

Offset: 1

R. H. Hardin, Oct 09 2011

Every 0 is next to 0 0's, every 1 is next to 1 0's, every 2 is next to 2 0's, every 3 is next to 3 0's, every 4 is next to 4 0's

Also, the number of maximal independent vertex sets in the grid graph P_n X P_k. - Andrew Howroyd, May 16 2017

			Table starts
..1...2....2.....3......4.......5........7.........9.........12..........16
..2...2....4.....6.....10......16.......26........42.........68.........110
..2...4...10....18.....38......78......156.......320........654........1326
..3...6...18....42....108.....274......692......1754.......4442.......11248
..4..10...38...108....358....1132.....3580.....11382......36270......114992
..5..16...78...274...1132....4468....17742.....70616.....281202.....1117442
..7..26..156...692...3580...17742....88056....439338....2192602....10912392
..9..42..320..1754..11382...70616...439338...2745186...17155374...106972582
.12..68..654..4442..36270..281202..2192602..17155374..134355866..1049189170
.16.110.1326.11248.114992.1117442.10912392.106972582.1049189170.10264692132
...
Some solutions for n=6 k=4
..0..2..1..0....0..2..0..1....2..0..2..0....0..3..0..2....0..2..1..0
..2..0..1..2....1..1..1..1....0..2..1..1....2..0..4..0....3..0..1..2
..1..1..2..0....1..0..2..0....2..1..0..2....1..2..0..2....0..3..1..0
..0..3..0..3....1..1..1..1....0..2..2..0....0..1..1..1....2..0..1..2
..3..0..4..0....0..3..0..2....3..0..1..2....1..1..1..0....1..1..2..0
..0..3..0..2....2..0..3..0....0..2..1..0....1..0..1..1....0..2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..364
Oh, Seungsang. "Maximal independent sets on a grid graph." Discrete Mathematics 340.12 (2017): 2762-2768. Also arXiv:1709.03678.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

Column 1 is A000931(n+6).
Column 2 is A006355(n+1).
Columns 3-7 are A197049, A197050, A197051, A197052, A197053.
Main diagonal is A197048.
Cf. A089934 (independent sets), A218354 (dominating sets).

A197149 Decimal expansion of the shortest distance from the x axis through (1,1) to the line y=3x.

Original entry on oeis.org

1, 9, 9, 9, 1, 5, 8, 3, 9, 9, 5, 8, 0, 3, 4, 4, 2, 6, 8, 8, 1, 7, 4, 2, 3, 5, 4, 3, 8, 4, 6, 1, 6, 4, 7, 3, 4, 1, 3, 2, 8, 2, 3, 4, 7, 1, 1, 8, 9, 0, 6, 8, 3, 7, 7, 0, 9, 2, 8, 1, 2, 8, 1, 0, 4, 6, 8, 5, 4, 8, 5, 2, 8, 7, 0, 0, 4, 9, 0, 3, 2, 7, 6, 5, 3, 7, 3, 7, 5, 1, 3, 1, 8, 9, 2, 0, 5, 5, 5

Offset: 1

Clark Kimberling, Oct 11 2011

The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T. For discussions and guides to related sequences, see A197032, A197008 and A195284.

			length of Philo line:    1.999158399580...
endpoint on x axis:    (1.60479, 0); see A197148
endpoint on line y=3x: (0.570212, 1.71064)

Cf. A197032, A197049, A197008, A195284.

f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
g[t_] := D[f[t], t]; Factor[g[t]]
p[t_] := h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 - 3 h m t^2 + m t^3
m = 3; h = 1; k = 1;(* slope m, point (h,k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t] (* A197148 *)
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t - m*h)],
N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=3x *)
d = N[Sqrt[f[t]], 100]
RealDigits[d]  (* A197149 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],
ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]

A231882 Number of maximal 2-independent sets in the planar 3 X n grid graph.

Original entry on oeis.org

0, 3, 4, 11, 17, 36, 69, 133, 254, 499, 959, 1852, 3589, 6943, 13410, 25951, 50197, 97050, 187699, 363047, 702066, 1357755, 2625947, 5078438, 9821417, 18994465, 36734648, 71043261, 137395463, 265718350, 513889567, 993844901, 1922062694, 3717202293, 7188941039

Offset: 0

N. J. A. Sloane, Nov 17 2013

nonn

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 (1,0,3,1,0,1,-2,0,-1).

Cf. A197054, A197049.

LinearRecurrence[{1,0,3,1,0,1,-2,0,-1},{0,3,4,11,17,36,69,133,254,499},40] (* Harvey P. Dale, Oct 05 2017 *)

Euler et al. give an explicit g.f. and recurrence.

G.f.: x*(3 + x + 7*x^2 - 3*x^3 + 4*x^4 - 4*x^5 - x^6 - 2*x^7 - x^8) / ((1 + x)*(1 - 2*x + 2*x^2 - 5*x^3 + 4*x^4 - 4*x^5 + 3*x^6 - x^7 + x^8)). - Colin Barker, Oct 03 2017

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

A293177 Triangle read by rows: T(m,n) = number of maximal independent sets in m X n rectangular grid graph (m>=1, 1<=n<=m).

Original entry on oeis.org

1, 2, 2, 2, 4, 10, 3, 6, 18, 42, 4, 10, 38, 108, 358, 5, 16, 78, 274, 1132, 4468, 7, 26, 156, 692, 3580, 17742, 88056, 9, 42, 320, 1754, 11382, 70616, 439338, 2745186, 12, 68, 654, 4442, 36270, 281202, 2192602, 17155374, 16, 110, 1326, 11248, 114992, 1117442, 10912392, 106972582

Offset: 1

N. J. A. Sloane, Oct 19 2017

			Triangle begins:
1,
2,2,
2,4,10,
3,6,18,42,
4,10,38,108,358,
5,16,78,274,1132,4468,
7,26,156,692,3580,17742,88056,
9,42,320,1754,11382,70616,439338,2745186,
12,68,654,4442,36270,281202,2192602,17155374,
16,110,1326,11248,114992,1117442,10912392,106972582,
...

Oh, Seungsang. "Maximal independent sets on a grid graph." Discrete Mathematics 340.12 (2017): 2762-2768. Also arXiv:1709.03678.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

Main diagonal is A197048.
Rows 3,4,5,6,... form the beginnings of A197049, A197050, A197051, A197052, ..., respectively.
Triangular version of A197054.

cp's OEIS Frontend

A197054 T(n,k)=Number of nXk 0..4 arrays with each element equal to the number of its horizontal and vertical zero neighbors.

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A197149 Decimal expansion of the shortest distance from the x axis through (1,1) to the line y=3x.

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Mathematica

A231882 Number of maximal 2-independent sets in the planar 3 X n grid graph.

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Programs

Mathematica

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A293177 Triangle read by rows: T(m,n) = number of maximal independent sets in m X n rectangular grid graph (m>=1, 1<=n<=m).

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