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

%I A320099 #20 Jun 09 2019 18:42:57
%S A320099 1,50,5193,583199,65485654,7354266811,825905301851,92751581627976,
%T A320099 10416273692997679,1169777980482365913,131369486228240893660,
%U A320099 14753177269494392259423,1656824927874469183283433,186066281959642930757881316,20895787297635543757965741097
%N A320099 Number of no-leaf subgraphs of the 5 X n grid.
%C A320099 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".
%H A320099 Peter Kagey, <a href="/A320099/b320099.txt">Table of n, a(n) for n = 1..488</a>
%F A320099 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.
%e A320099 Three of the a(3) = 5193 subgraphs of the 5 X 3 grid with no leaf vertices are:
%e A320099 +---+---+      +   +   +      +   +---+
%e A320099 |   |   |                         |   |
%e A320099 +---+---+      +---+---+      +   +---+
%e A320099 |   |   |,     |   |   |, and          .
%e A320099 +---+---+      +   +---+      +---+   +
%e A320099 |   |   |      |   |          |   |
%e A320099 +---+---+      +---+   +      +---+---+
%e A320099 |   |   |                         |   |
%e A320099 +---+---+      +   +   +      +   +---+
%Y A320099 A093129 is analogous for 2 X (n+1) grids.
%Y A320099 A301976 is analogous for 3 X n grids.
%Y A320099 A320097 is analogous for 4 X n grids.
%K A320099 nonn
%O A320099 1,2
%A A320099 _Peter Kagey_, Oct 05 2018