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%I #31 Apr 02 2020 14:06:07
%S 1,10,101,1105,12046,131399,1433341,15635350,170555501,1860475165,
%T 20294671306,221380909199,2414895329881,26342467719490,
%U 287352249584501,3134532277710025,34192502805225766,372982998579773399,4068620481572281621,44381842298715324430,484131644804296287101
%N Number of self-avoiding paths in (2*n+1) X 3 grid starting the upper left corner, passing through the center of grid and finishing the lower right corner.
%H Seiichi Manyama, <a href="/A333686/b333686.txt">Table of n, a(n) for n = 0..500</a>
%F Conjecture: G.f.: (1-3*x-x^2)*(1+3*x+x^2+x^3)/((1-x)*(1+x)*(1+x+x^2)*(1-11*x+x^2)).
%F Conjecture: a(n) = 10*a(n-1) + 10*a(n-2) - 10*a(n-4) - 10*a(n-5) + a(n-6) for n>5.
%e a(0) = 1;
%e S--+--E
%e a(1) = 10;
%e S--*--* S--*--* S--* S--* S--*
%e | | | | |
%e +--* *--+--* +--* + *--+
%e | | | | |
%e *--E *--*--E E *--E *--*--E
%e S *--* S *--* S S S
%e | | | | | | | | |
%e * + * *--+ * * +--* *--+--* *--+
%e | | | | | | | | |
%e *--* E E *--* E E *--E
%e a(2) = 101;
%e S--*--* S--*--* S--*--* S--*--* S--*--*
%e | | | | |
%e *--*--* *--*--* *--*--* *--*--* *--*--*
%e | | | | |
%e *--+--* *--+ *--+ *--+ * +--*
%e | | | | | | |
%e * *--* *--* * *--* *
%e | | | | |
%e E *--*--E E *--E E
%e ... and so on.
%o (Python)
%o # Using graphillion
%o from graphillion import GraphSet
%o import graphillion.tutorial as tl
%o def A333685(n, k):
%o if n == 0 or k == 0: return 1
%o universe = tl.grid(2 * n, 2 * k)
%o GraphSet.set_universe(universe)
%o start, goal = 1, (2 * n + 1) * (2 * k + 1)
%o paths = GraphSet.paths(start, goal).including((start + goal) // 2)
%o return paths.len()
%o def A333686(n):
%o return A333685(n, 1)
%o print([A333686(n) for n in range(20)])
%Y Column 1 of A333685.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Apr 02 2020