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A365970 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.

%I A365970 #30 Jun 10 2025 13:54:23
%S A365970 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 A365970 169,727,276,11,0,0,0,1,285,3223,3440,432,4,0,0,0,1,356,10853,27632,
%U A365970 10141,459,2,0,0,0,0,344,27198,155524,134527,19597,314,0,0,0
%N A365970 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 A365970 Polyforms are "free" in that they are counted up to rotation and reflection.
%C A365970 Conjecture: Columns and antidiagonals are unimodal.
%C A365970 Rows sums are given by A343909.
%H A365970 Bert Dobbelaere, <a href="/A365970/b365970.txt">Table of n, a(n) for n = 0..152</a>
%H A365970 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 A365970 Peter Kagey, <a href="/A365970/a365970.txt">Haskell program</a>.
%H A365970 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>.
%F A365970 T(n,k) = 0 for k > n - floor((n - 1)/4).
%e A365970 Triangle begins:
%e A365970   1;
%e A365970   1, 1;
%e A365970   0, 1,   0;
%e A365970   0, 3,   1,     0;
%e A365970   0, 3,   5,     1,      0;
%e A365970   0, 6,  24,    13,      1,      0;
%e A365970   0, 3,  74,   105,     13,      0,     0;
%e A365970   0, 3, 169,   727,    276,     11,     0,   0;
%e A365970   0, 1, 285,  3223,   3440,    432,     4,   0, 0;
%e A365970   0, 1, 356, 10853,  27632,  10141,   459,   2, 0, 0;
%e A365970   0, 0, 344, 27198, 155524, 134527, 19597, 314, 0, 0, 0.
%Y A365970 Cf. A343909.
%K A365970 nonn,tabl,hard
%O A365970 0,8
%A A365970 _Peter Kagey_, Sep 23 2023