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%I #24 Sep 25 2023 08:23:12
%S 1,1,1,0,1,0,0,3,1,0,0,3,5,1,0,0,6,24,13,1,0,0,3,74,105,13,0,0,0,3,
%T 169,727,276,11,0,0,0,1,285,3223,3440,432,4,0,0,0,1,356,10853,27632,
%U 10141,459,2,0,0,0,0,344,27198,155524,134527,19597,314,0,0,0
%N Triangle read by rows: T(n,k) is the number of generalized polyforms on the tetrahedral-octahedral honeycomb with n cells, k of which are octahedra; 0 <= k <= n.
%C Polyforms are "free" in that they are counted up to rotation and reflection.
%C Conjecture: Columns and antidiagonals are unimodal.
%C Rows sums are given by A343909.
%H Peter Kagey, <a href="/A343909/a343909.gif">Animation of the A343909(4) = 9 polyforms with 4 cells and T(4,1) = 3, T(4,2) = 5, and T(4,3) = 1 octahedra</a>.
%H Math Stack Exchange, <a href="https://math.stackexchange.com/q/4128528/121988">Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb</a>.
%H Peter Kagey, <a href="/A365970/a365970.txt">Haskell program</a>.
%F T(n,k) = 0 for k > n - floor((n - 1)/4).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 0, 1, 0;
%e 0, 3, 1, 0;
%e 0, 3, 5, 1, 0;
%e 0, 6, 24, 13, 1, 0;
%e 0, 3, 74, 105, 13, 0, 0;
%e 0, 3, 169, 727, 276, 11, 0, 0;
%e 0, 1, 285, 3223, 3440, 432, 4, 0, 0;
%e 0, 1, 356, 10853, 27632, 10141, 459, 2, 0, 0;
%e 0, 0, 344, 27198, 155524, 134527, 19597, 314, 0, 0, 0.
%Y Cf. A343909.
%K nonn,tabl,hard
%O 0,8
%A _Peter Kagey_, Sep 23 2023