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A189604 Number of n X 3 array permutations with each element not moving, or moving one space E, S or NW.

%I A189604 #20 Feb 08 2022 22:26:16
%S A189604 1,6,20,72,256,912,3248,11568,41200,146736,522608,1861296,6629104,
%T A189604 23609904,84087920,299483568,1066626544,3798846768,13529793392,
%U A189604 48187073712,171620807920,611236571184,2176951329392,7753327130544
%N A189604 Number of n X 3 array permutations with each element not moving, or moving one space E, S or NW.
%C A189604 Column 3 of A189610.
%C A189604 Binomial transform of A006131 starting (1, 5, 9, 29, 65, ...). - _Gary W. Adamson_, Feb 19 2014
%H A189604 R. H. Hardin, <a href="/A189604/b189604.txt">Table of n, a(n) for n = 1..200</a>
%F A189604 Empirical: a(n) = 3*a(n-1) + 2*a(n-2).
%F A189604 G.f.: (x+3*x^2)/(1-3*x-2*x^2). - _Vladimir Kruchinin_, May 13 2011
%e A189604 Some solutions for 4 X 3:
%e A189604 .
%e A189604    4  5  1    0  5  1    0  1  2    0  1  2
%e A189604    0  3  2    7  4  2    3  4  5    3  4  5
%e A189604    6  7  8    3  6  8    6 11  8   10  7  8
%e A189604    9 10 11    9 10 11    9  7 10    6  9 11
%e A189604 .
%e A189604    4  0  1    0  1  2    4  1  2
%e A189604    7  3  2    3  8  5    0  3  5
%e A189604   10 11  5    6  4  7    6  7  8
%e A189604    6  9  8    9 10 11    9 10 11
%t A189604 a[n_] := Sum[Sum[4^j Binomial[k-j+1, j], {j, 0, Quotient[k+1, 2]}]* Binomial[n-1, k], {k, 0, n-1}];
%t A189604 a /@ Range[1, 24] (* _Jean-François Alcover_, Sep 24 2019, after _Gary W. Adamson_ *)
%Y A189604 Cf. A006131.
%K A189604 nonn
%O A189604 1,2
%A A189604 _R. H. Hardin_, Apr 24 2011