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%I #30 Aug 20 2015 23:03:06
%S 1,2,3,3,1,2,1,2,3,1,2,3,1,2,3,3,1,2,3,1,2,1,2,3,1,2,3,1,2,3,1,2,3,1,
%T 2,3,3,1,2,3,1,2,3,1,2,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,3,1,
%U 2,3,1,2,3,1,2,3,1,2,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3
%N Matrix given by f(i,j) = 1 + [(2i+j) mod 3], by antidiagonals.
%C This data is used to specify the height of hexagonally packed cylinders in a triangle with open boundaries. Three cylinders that touch each other define a "triple" and water can be retained between these cylinders. A257594, A258445 and A259052 give a classification for such spaces. The links below ignore the inter-cylinder space retention and only consider the water retention above solid cylinders. - _Craig Knecht_, Jul 16 2015
%H Craig Knecht, <a href="/A204259/a204259.jpg">Water retention triple.</a>
%H Craig Knecht, <a href="/A204259/a204259_1.jpg">Row sums of numbers completely surrounded by larger numbers (water retention) in A204259 = A008611.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Water_retention_on_mathematical_surfaces">Water retention on mathematical surfaces</a>
%e Northwest corner:
%e 1 2 3 1 2 3 1 2
%e 3 1 2 3 1 2 3 1
%e 2 3 1 2 3 1 2 3
%e 1 2 3 1 2 3 1 2
%e 3 1 2 3 1 2 3 1
%e 2 3 1 2 3 1 2 3
%t f[i_, j_] := 1 + Mod[2 i + j, 3];
%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t TableForm[m[8]] (* 8x8 principal submatrix *)
%t Flatten[Table[f[i, n + 1 - i],
%t {n, 1, 12}, {i, 1, n}]] (* A204259 *)
%t Permanent[m_] :=
%t With[{a = Array[x, Length[m]]},
%t Coefficient[Times @@ (m.a), Times @@ a]];
%t Table[Permanent[m[n]], {n, 1, 20}] (* A204258 *)
%Y Cf. A204260.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jan 14 2012
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