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A349596 Number of edge subsets E of the 3 X n grid graph such that E contains a path between the top left node and the bottom right node.

%I A349596 #43 Nov 20 2024 07:07:56
%S A349596 1,40,1135,28942,707239,16963938,403490839,9560192914,226115020735,
%T A349596 5343643837642,126235739481031,2981618243157330,70418570359871599,
%U A349596 1663054542669694138,39275207266744385815,927528207559891996258,21904544495171662611391,517297785739589326153482
%N A349596 Number of edge subsets E of the 3 X n grid graph such that E contains a path between the top left node and the bottom right node.
%C A349596 a(n)/2^(5*n-3) is the probability that the top left and bottom right vertices of the 3 X n grid graph are still connected after each edge has been independently deleted with probability 1/2. - _Pontus von Brömssen_, May 25 2024
%H A349596 Pontus von Brömssen, <a href="/A349596/b349596.txt">Table of n, a(n) for n = 1..728</a>
%H A349596 Steven B Segletes, <a href="https://apps.dtic.mil/sti/citations/AD1090614">On the Electrical Connectivity of a 2-D, Randomly Distributed, Two-Component (Conducting/Insulating) Mixture</a>, page 12 lists number for 3 X 3 maze.
%H A349596 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (54,-1077,10642,-57954,180960,-324992,325632,-165888,32768).
%F A349596 a(n) = 54*a(n-1) - 1077*a(n-2) + 10642*a(n-3) - 57954*a(n-4) + 180960*a(n-5) - 324992*a(n-6) + 325632*a(n-7) - 165888*a(n-8) + 32768*a(n-9) for n >= 10. - _Pontus von Brömssen_, May 25 2024
%F A349596 G.f.: (1 - 14*x + 52*x^2 + 90*x^3 - 960*x^4 + 2096*x^5 - 1792*x^6 + 512*x^7)/((1 - 15*x + 48*x^2 - 32*x^3)*(1 - 39*x + 444*x^2 - 2078*x^3 + 4224*x^4 - 3648*x^5 + 1024*x^6)). - _Eugene Nonko_, Nov 15 2024
%Y A349596 Cf. A349594.
%Y A349596 Third row/column of A373036.
%K A349596 nonn,easy
%O A349596 1,2
%A A349596 _Eugene Nonko_, Nov 22 2021
%E A349596 a(11)-a(18) from _Martin Ehrenstein_, Dec 13 2021
%E A349596 Name clarified by _Eugene Nonko_, Nov 18 2024