A320097 -id:A320097 - cp's OEIS frontend

A320099 Number of no-leaf subgraphs of the 5 X n grid.

1, 50, 5193, 583199, 65485654, 7354266811, 825905301851, 92751581627976, 10416273692997679, 1169777980482365913, 131369486228240893660, 14753177269494392259423, 1656824927874469183283433, 186066281959642930757881316, 20895787297635543757965741097

Offset: 1

Peter Kagey, Oct 05 2018

nonn

Also, the number of ways to lay unit-length matchsticks on a 5 X n grid of points in such a way that no end is "orphaned".

			Three of the a(3) = 5193 subgraphs of the 5 X 3 grid with no leaf vertices are:
+---+---+      +   +   +      +   +---+
|   |   |                         |   |
+---+---+      +---+---+      +   +---+
|   |   |,     |   |   |, and          .
+---+---+      +   +---+      +---+   +
|   |   |      |   |          |   |
+---+---+      +---+   +      +---+---+
|   |   |                         |   |
+---+---+      +   +   +      +   +---+

Peter Kagey, Table of n, a(n) for n = 1..488

A093129 is analogous for 2 X (n+1) grids.
A301976 is analogous for 3 X n grids.
A320097 is analogous for 4 X n grids.

Conjecture: a(n) = 103*a(n-1) + 1063*a(n-2) - 1873*a(n-3) - 20274*a(n-4) + 44071*a(n-5) - 10365*a(n-6) - 20208*a(n-7) + 5959*a(n-8) + 2300*a(n-9) - 500*a(n-10) for n > 10.

A320101 Table read by rows: T(n,k) is the number of no-leaf subgraphs of the n X k grid where 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 5, 43, 1, 15, 463, 16372, 1, 50, 5193, 583199, 65485654, 1, 175, 58653, 20788249, 7354266811, 2602065897364, 1, 625, 663203, 741026781, 825905301851

Offset: 1

Peter Kagey, Oct 05 2018

			Three of the T(4,3) = 463 subgraphs of the 4 X 3 grid with no leaf vertices are
  +   +---+      +   +   +      +   +---+
      |   |                         |   |
  +---+---+      +---+---+      +   +---+
  |   |    ,     |   |   |, and          .
  +---+   +      +   +---+      +---+   +
  |   |          |   |          |   |
  +---+   +      +---+   +      +---+   +
Table begins:
  n\k|    1    2     3        4          5             6
  ---+---------------------------------------------------
   1 |    1
   2 |    1    2
   3 |    1    5    43
   4 |    1   15   463    16372
   5 |    1   50  5193   583199   65485654
   6 |    1  175 58653 20788249 7354266811 2602065897364

Cf. A093129, A301976, A320097, A320099.

cp's OEIS Frontend

A320099 Number of no-leaf subgraphs of the 5 X n grid.

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