A288029 -id:A288029 - cp's OEIS frontend

A288025 Array read by antidiagonals: T(m,n) = number of minimal edge covers in the grid graph P_m X P_n.

0, 1, 1, 1, 2, 1, 1, 6, 6, 1, 2, 17, 38, 17, 2, 2, 45, 190, 190, 45, 2, 3, 120, 1021, 1834, 1021, 120, 3, 4, 324, 5494, 19988, 19988, 5494, 324, 4, 5, 873, 29042, 208186, 419710, 208186, 29042, 873, 5, 7, 2349, 154772, 2177591, 8704085, 8704085, 2177591, 154772, 2349, 7

Offset: 1

Andrew Howroyd, Jun 04 2017

A minimal edge cover is an edge cover such that the removal of any edge in the cover destroys the covering property. Equivalently, these are the edge covers whose connected components are stars. A minimal edge cover is not the same as a minimum edge cover.

			Table starts:
================================================================
m\n| 1   2     3       4         5           6             7
---|------------------------------------------------------------
1  | 0   1     1       1         2           2             3 ...
2  | 1   2     6      17        45         120           324 ...
3  | 1   6    38     190      1021        5494         29042 ...
4  | 1  17   190    1834     19988      208186       2177591 ...
5  | 2  45  1021   19988    419710     8704085     179649371 ...
6  | 2 120  5494  208186   8704085   356269056   14484264119 ...
7  | 3 324 29042 2177591 179649371 14484264119 1163645044100 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Minimal Edge Cover

Main diagonal is A288027.
Rows 1-3 are A182097, A288029, A288030.
Cf. A286912.

A288030 Number of minimal edge covers in the grid graph P_3 X P_n.

Original entry on oeis.org

1, 6, 38, 190, 1021, 5494, 29042, 154772, 824695, 4386942, 23356322, 124344111, 661873859, 3523418150, 18756407661, 99845472493, 531509598443, 2829393615575, 15061734439895, 80178331797652, 426814323111204, 2272065745374039, 12094915689644237

Offset: 1

Andrew Howroyd, Jun 04 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Minimal Edge Cover.

Row 3 of A288025.

Cf. A288029, A288027.

Empirical: a(n) = 3*a(n-1)+6*a(n-2)+35*a(n-3) -7*a(n-4)+15*a(n-5)-55*a(n-6) +31*a(n-7)+36*a(n-8)+36*a(n-9) -124*a(n-10)-8*a(n-12)+32*a(n-13) +a(n-14)+9*a(n-15)-2*a(n-16) -2*a(n-17)-a(n-18) for n>18.

Empirical g.f.: x*(1 + 3*x + 14*x^2 + 5*x^3 + 20*x^4 - 12*x^5 + 15*x^6 + 6*x^7 + 2*x^8 - 82*x^9 - 5*x^10 - 3*x^11 + 29*x^12 + 8*x^14 - 2*x^15 - 2*x^16 - x^17) / (1 - 3*x - 6*x^2 - 35*x^3 + 7*x^4 - 15*x^5 + 55*x^6 - 31*x^7 - 36*x^8 - 36*x^9 + 124*x^10 + 8*x^12 - 32*x^13 - x^14 - 9*x^15 + 2*x^16 + 2*x^17 + x^18). - Colin Barker, Jun 16 2017

cp's OEIS Frontend

A288025 Array read by antidiagonals: T(m,n) = number of minimal edge covers in the grid graph P_m X P_n.

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