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 A301976 Number of no-leaf subgraphs of the 3 X n grid. 5

%I #21 Oct 06 2018 04:14:59

%S 1,5,43,463,5193,58653,663203,7500343,84825873,959351093,10849935003,

%T 122709094303,1387798370393,15695530423373,177511143297043,

%U 2007591024144903,22705175829637153,256787863292718693,2904183928335418123,32845338488555237743

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

%C 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".

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

%H Peter Kagey, <a href="/A301976/b301976.txt">Table of n, a(n) for n = 1..949</a>

%F Conjectures from _Colin Barker_, Mar 30 2018: (Start)

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

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

%F (End)

%e Three of the a(4) = 463 subgraphs of the 3 X 4 grid with no leaf vertices are

%e +---+ +---+ + + +---+ + + +---+

%e | | | | | | | |

%e +---+---+ +, + +---+---+, and +---+ +---+.

%e | | | | | | |

%e +---+---+---+ + +---+ + +---+ + +

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

%K nonn

%O 1,2

%A _Peter Kagey_, Mar 29 2018

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