A286911 -id:A286911 - cp's OEIS frontend

A286912 Array read by antidiagonals: T(m,n) = number of edge covers in the grid graph P_m X P_n.

0, 1, 1, 1, 7, 1, 2, 43, 43, 2, 3, 277, 969, 277, 3, 5, 1777, 23663, 23663, 1777, 5, 8, 11407, 571099, 2180738, 571099, 11407, 8, 13, 73219, 13807469, 198906617, 198906617, 13807469, 73219, 13, 21, 469981, 333735575, 18169793971, 68534828391, 18169793971, 333735575, 469981, 21

Offset: 1

Andrew Howroyd, May 15 2017

			Table starts:
======================================================================
m\n| 1     2        3           4              5                 6
---|------------------------------------------------------------------
1  | 0     1        1           2              3                 5 ...
2  | 1     7       43         277           1777             11407 ...
3  | 1    43      969       23663         571099          13807469 ...
4  | 2   277    23663     2180738      198906617       18169793971 ...
5  | 3  1777   571099   198906617    68534828391    23650967140325 ...
6  | 5 11407 13807469 18169793971 23650967140325 30833670159649637 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..276
Eric Weisstein's World of Mathematics, Edge Cover
Eric Weisstein's World of Mathematics, Grid Graph

Rows 1-3 are A000045(n-1), A286911, A288031.
Main diagonal is A286913.
Cf. A210662, A099390, A089934, A218354.

T(1,1) corrected by Andrew Howroyd, Jun 04 2017

A286913 Number of edge covers in the grid graph P_n X P_n.

Original entry on oeis.org

0, 7, 969, 2180738, 68534828391, 30833670159649637, 197887615273032627789510, 18126687290150589819559507400227, 23696879029605485832353513435527035363501, 442121584517675331278913696274915728729945474905362

Offset: 1

Andrew Howroyd, May 15 2017

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..12
Eric Weisstein's World of Mathematics, Edge Cover
Eric Weisstein's World of Mathematics, Grid Graph

Main diagonal of A286912.

Cf. A286911.

A286912 = Cases[Import["https://oeis.org/A286912/b286912.txt", "Table"], {, }][[All, 2]];
a[n_] := A286912[[2 n^2 - 2 n + 1]]; a[1] = 1;
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Sep 23 2019 *)

a(1) corrected by Andrew Howroyd, Jan 29 2023

A286945 Number of maximal matchings in the ladder graph P_2 X P_n.

Original entry on oeis.org

1, 2, 5, 11, 24, 51, 109, 234, 503, 1081, 2322, 4987, 10711, 23006, 49415, 106139, 227976, 489669, 1051759, 2259072, 4852259, 10422163, 22385754, 48082339, 103276009, 221826440, 476460797, 1023389687, 2198137722, 4721377893, 10141043023, 21781936530

Offset: 1

Andrew Howroyd, May 16 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500
Svenja Huntemann, Neil A. McKay, Counting Domineering Positions, arXiv:1909.12419 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Ladder Graph
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Maximal Independent Edge Set
Index entries for linear recurrences with constant coefficients, signature (2,0,0,1,1).

Cf. A284703, A284710, A286911.

a:=[1,2,5,11,24];; for n in [6..35] do a[n]:=2*a[n-1]+a[n-4]+a[n-5]; od; a; # G. C. Greubel, Dec 30 2019

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( x*(1+x^2+x^3+x^4)/(1-2*x-x^4-x^5) )); // G. C. Greubel, Dec 30 2019

seq(coeff(series(x*(1+x^2+x^3+x^4)/(1-2*x-x^4-x^5), x, n+1), x, n), n = 1..35); # G. C. Greubel, Dec 30 2019

Table[3Cos[nPi/3]/13 - 5Sin[nPi/3]/(13 Sqrt[3]) + RootSum[-1 -2# -#^2 +#^3 &, (-6 -72# +80#^2) #^n &]/403, {n, 35}] (* Eric W. Weisstein, Jul 13 2017 *)
LinearRecurrence[{2,0,0,1,1}, {1,2,5,11,24}, 35] (* Eric W. Weisstein, Jul 13 2017 *)
CoefficientList[Series[(1+x^2+x^3+x^4)/(1-2x-x^4-x^5), {x, 0, 35}], x] (* Eric W. Weisstein, Jul 13 2017 *)

Vec((1+x^2+x^3+x^4)/((1-x+x^2)*(1-x-2*x^2-x^3)) + O(x^35))

def A286945_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x^2+x^3+x^4)/(1-2*x-x^4-x^5) ).list()
a=A286945_list(35); a[1:] # G. C. Greubel, Dec 30 2019

a(n) = 2*a(n-1) + a(n-4) + a(n-5) for n>5.

G.f.: x*(1+x^2+x^3+x^4)/((1-x+x^2)*(1-x-2*x^2-x^3)).

A288031 Number of edge covers in the grid graph P_3 X P_n.

Original entry on oeis.org

1, 43, 969, 23663, 571099, 13807469, 333735575, 8066926825, 194989463233, 4713185791699, 113924706164937, 2753729539353359, 66561737202707371, 1608896152717277333, 38889412128248718215, 940014912175876488361, 22721558047666401897553, 549213840574693856578267

Offset: 1

Andrew Howroyd, Jun 04 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row 3 of A286912.

Cf. A286911, A286913.

Empirical: a(n) = 20*a(n-1)+100*a(n-2)+24*a(n-3) -95*a(n-4)+10*a(n-5)+8*a(n-6) for n>6.

Empirical g.f.: x*(1 + 23*x + 9*x^2 - 41*x^3 + 2*x^4 + 8*x^5) / (1 - 20*x - 100*x^2 - 24*x^3 + 95*x^4 - 10*x^5 - 8*x^6). - Colin Barker, Jun 11 2017

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A286912 Array read by antidiagonals: T(m,n) = number of edge covers in the grid graph P_m X P_n.

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A286913 Number of edge covers in the grid graph P_n X P_n.

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A286945 Number of maximal matchings in the ladder graph P_2 X P_n.

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A288031 Number of edge covers in the grid graph P_3 X P_n.

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