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Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.
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%I #43 May 18 2022 11:50:40

%S 1,2,2,6,8,22,42,112,252,650,1584,4091,10369,26938,69651,182116,

%T 476272,1253067,3302187,8733551,23142116,61477564,163612714,436278921,

%U 1165218495,3117021788

%N Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

%C See the link below for a definition of the tetrakis square tiling. When a square grid cell is divided into triangles, it must be divided dexter (\) or sinister (/) according to the parity of the grid cell.

%D Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

%H Peter Kagey, <a href="/A197465/a197465.pdf">Example illustrating a(4) = 6</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetrakis_square_tiling">Tetrakis square tiling</a>

%e For n=3 there are 4 triaboloes. Of these, 2 conform to the tetrakis grid. Each of these 2 has a unique dissection into 6 tetrakis cells. - _George Sicherman_, Mar 25 2021

%Y Cf. A197466, A197467, A006074.

%Y Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

%K nonn,hard,more

%O 1,2

%A _Joseph Myers_, Oct 15 2011

%E Name clarified by _George Sicherman_, Mar 25 2021

%E a(21)-a(26) from _Aaron N. Siegel_, May 18 2022