This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A197459 #51 Jun 04 2025 15:57:44 %S A197459 1,1,3,4,12,27,78,208,635,1859,5726,17526,54620,170479,536714,1694567, %T A197459 5376764,17110286,54631302,174879997,561229678,1805022806,5817191196, %U A197459 18781911278,60744460580 %N A197459 Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant). %C A197459 [3.6.3.6] refers to the face configuration of the rhombille tiling. - _Peter Kagey_, Mar 01 2020 %C A197459 If we draw the short diagonals of each tile in the rhombille tiling, we get a subset of edges of the regular hexagonal grid; two edges are adjacent if and only if the corresponding rhombi are adjacent. These are polyedges where angles are constrained to 120 degrees. So there is a 1-to-1 correspondence with the subset of polyedges counted in A159867 after removing polyedges with angles of 60 and/or 180 degrees. - _Joseph Myers_, Jul 12 2020 %C A197459 These are also known as polytwigs, by association with their representation as polyedges. - _Aaron N. Siegel_, May 15 2022 %D A197459 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4. %H A197459 Abaroth's World, <a href="http://abarothsworld.com/Puzzles/Polyforms/Polytwigs.htm">Illustration of a(1) through a(6)</a> %H A197459 Peter Kagey, <a href="/A197459/a197459.pdf">Illustration for A197459(4) = 4</a>. %H A197459 R. J. Mathar, <a href="/A197459/a197459_1.pdf">OEIS A197459</a> %H A197459 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rhombille_tiling">Rhombille tiling</a> %Y A197459 Cf. A019988, A159867, A197460, A197461. %Y A197459 Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square). %K A197459 hard,nonn,more %O A197459 1,3 %A A197459 _Joseph Myers_, Oct 15 2011 %E A197459 a(18)-a(22) from _Aaron N. Siegel_, May 15 2022 %E A197459 a(23)-a(25) from _Bert Dobbelaere_, Jun 04 2025