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A320097 Number of no-leaf subgraphs of the 4 X n grid.

%I A320097 #21 Oct 20 2018 04:00:14
%S A320097 1,15,463,16372,583199,20788249,741026781,26415034787,941604528692,
%T A320097 33564941612743,1196473967526971,42650154782713601,
%U A320097 1520330364358307239,54194514148101568538,1931846809485041315873,68863650758427752078777,2454750745501814744040599
%N A320097 Number of no-leaf subgraphs of the 4 X n grid.
%C A320097 Also, the number of ways to lay unit-length matchsticks on a 4 X n grid of points in such a way that no end is "orphaned".
%H A320097 Peter Kagey, <a href="/A320097/b320097.txt">Table of n, a(n) for n = 1..500</a>
%F A320097 Conjecture: a(n) = 36*a(n-1) - 7*a(n-2) - 201*a(n-3) + 49*a(n-4) + 20*a(n-5) - 5*a(n-6) for all n > 6.
%F A320097 Empirical g.f.: x*(1 - 21*x - 70*x^2 + 10*x^3 + 14*x^4 - 3*x^5) / (1 - 36*x + 7*x^2 + 201*x^3 - 49*x^4 - 20*x^5 + 5*x^6). - _Colin Barker_, Oct 20 2018
%e A320097 Three of the a(3) = 463 subgraphs of the 4 X 3 grid with no leaf vertices are
%e A320097   +   +---+          +   +   +          +   +---+
%e A320097       |   |                                 |   |
%e A320097   +---+---+          +---+---+          +   +---+
%e A320097   |   |    ,         |   |   |,   and            .
%e A320097   +---+   +          +   +---+          +---+   +
%e A320097   |   |              |   |              |   |
%e A320097   +---+   +          +---+   +          +---+   +
%Y A320097 A093129 is analogous for 2 X (n+1) grids.
%Y A320097 A301976 is analogous for 3 X n grids.
%K A320097 nonn
%O A320097 1,2
%A A320097 _Peter Kagey_, Oct 05 2018