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