A301976 Number of no-leaf subgraphs of the 3 X n grid.

1, 5, 43, 463, 5193, 58653, 663203, 7500343, 84825873, 959351093, 10849935003, 122709094303, 1387798370393, 15695530423373, 177511143297043, 2007591024144903, 22705175829637153, 256787863292718693, 2904183928335418123, 32845338488555237743

Offset: 1

Author

Peter Kagey, Mar 29 2018

Keywords

• nonn

Comments

Also, the number of ways to lay unit-length matchsticks on a 3 X n grid of points in such a way that no end is "orphaned".

Conjecture: a(n) mod 10 = 3 for n > 2.

Examples

			Three of the a(4) = 463 subgraphs of the 3 X 4 grid with no leaf vertices are
  +---+   +---+      +   +   +---+      +   +   +---+
  |   |   |   |              |   |              |   |
  +---+---+   +,     +   +---+---+, and +---+   +---+.
  |   |       |          |   |          |   |
  +---+---+---+      +   +---+   +      +---+   +   +
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..949

Crossrefs

A093129 is analogous for 2 X (n+1) grids.

Formula

Conjectures from Colin Barker, Mar 30 2018: (Start)

G.f.: x*(1 + x)*(1 - 8*x - 3*x^2) / (1 - 12*x + 6*x^2 + 20*x^3 + 5*x^4).

a(n) = 12*a(n-1) - 6*a(n-2) - 20*a(n-3) - 5*a(n-4) for n>4.

(End)