cp's OEIS Frontend

A385390 Irregular triangle read by rows: T(n,k) is the number of polysticks of size k, i.e., connected subsets of k edges, of the n X n flat torus, up to cyclic shifts and reflections of rows and columns, as well as interchange of rows and columns; 1 <= k <= 2*n^2.

%I A385390 #7 Jun 29 2025 11:07:56
%S A385390 1,1,1,2,3,7,4,4,1,1,1,2,5,14,38,111,261,500,654,648,486,305,144,61,
%T A385390 19,6,1,1,1,2,5,16,52,199,759,2921,10668,36761,115231,322237,778242,
%U A385390 1576259,2591721,3412285,3671098,3320276,2565917,1717088,996355,503860,220074,83408,26783,7438,1678,351,52,11,1,1
%N A385390 Irregular triangle read by rows: T(n,k) is the number of polysticks of size k, i.e., connected subsets of k edges, of the n X n flat torus, up to cyclic shifts and reflections of rows and columns, as well as interchange of rows and columns; 1 <= k <= 2*n^2.
%C A385390 For n = 4, there are 384 automorphisms of (the line graph of) the 4 X 4 torus grid graph (it is isomorphic to the 4-dimensional hypercube graph), but here we only consider the subgroup consisting of the 128 symmetries of the 4 X 4 torus. Using the full automorphism group of the torus grid graph would change row 4 to the corresponding row of A333333.
%F A385390 T(n,k) = A019988(k) if n >= k.
%F A385390 T(n,k) >= A385388(n,k)/2, with equality if and only if k is odd.
%e A385390 Triangle begins:
%e A385390   1, 1;
%e A385390   1, 2, 3,  7,  4,   4,   1,   1;
%e A385390   1, 2, 5, 14, 38, 111, 261, 500, 654, 648, 486, 305, 144, 61, 19, 6, 1, 1;
%e A385390   ...
%Y A385390 Cf. A019988, A333333, A385385 (polyominoes), A385388 (interchange of rows and columns of the torus not allowed), A385389 (row sums).
%K A385390 nonn,tabf
%O A385390 1,4
%A A385390 _Pontus von Brömssen_, Jun 27 2025