A397023 - OEIS

%I #9 Jun 24 2026 23:18:39

%S 0,1,7,55,455,4191,39864,390534,3884896,39149945,398346975,4085997296

%N Number of free polynars with n cells that are not the 2 X enlargement of a free n-omino.

%C A390620(n) counts the free polynars with n cells, equivalently the n-fold multiples of the O-tetromino (edge-connected unions of n axis-aligned 2 X 2 blocks glued at integer offsets). Those whose blocks all lie on a common even sublattice are exactly the 2 X enlargements 2P of the free n-ominoes P, and there are A000105(n) of these; this sequence counts the rest. Equivalently, a(n) is the number of free polynars with n cells in which every tiling by 2 X 2 blocks uses at least one half-edge (one-step offset) gluing. See A397019 for the related family of tetromino multiples.

%H George Sicherman, <a href="https://sicherman.net/polynar/index.html">Catalogue of Polynars</a>.

%H Haoyang Xu, <a href="/A397023/a397023.png">Illustration of a(2)</a>

%H <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.

%F a(n) = A390620(n) - A000105(n).

%e a(2) = 1: realized as multiples of the O-tetromino, the two free polynars with 2 cells are two 2 X 2 blocks meeting along a full edge (the 2 X 4 rectangle, = 2 X the domino) and two blocks meeting along a half edge (one offset octomino); only the latter is not a 2 X enlargement, so a(2) = 1.

%e a(3) = A390620(3) - A000105(3) = 9 - 2 = 7.

%Y Cf. A390620 (free polynars), A000105 (free polyominoes), A390621 (one-sided polynars), A390622 (fixed polynars), A397019, A397020, A397021, A397022.

%K nonn,more,new

%O 1,3

%A _Haoyang Xu_, Jun 19 2026