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A061348 Consider a (solid) triangle with n cells on each edge, for a total of n(n+1)/2 cells; a(n) is number of inequivalent ways of labeling cells with 0's and 1's; triangle may be rotated and turned over.

%I A061348 #24 Aug 17 2024 16:52:33
%S A061348 2,4,20,208,5728,351616,44772352,11453771776,5864078802944,
%T A061348 6004800040206336,12297829416834170880,50371909152808594571264,
%U A061348 412646679762074900658913280,6760803201217259503457555972096,221537999297485988040673580072042496
%N A061348 Consider a (solid) triangle with n cells on each edge, for a total of n(n+1)/2 cells; a(n) is number of inequivalent ways of labeling cells with 0's and 1's; triangle may be rotated and turned over.
%F A061348 See Maple code for formula.
%e A061348 a(2) = 4, the labelings being {000}, {001}, {011}, {111}. Some of the 20 solutions for n=3 are as follows:
%e A061348 ..0......1.......0......1.......1.......1.......0
%e A061348 .0.0....0.0.....1.0....1.0.....0.0.....0.0.....1.1
%e A061348 0.0.0..0.0.0...0.0.0..0.0.0...1.0.0...0.1.0...0.0.0
%e A061348 The first solution for n = 4 is
%e A061348 ...0
%e A061348 ..0.0
%e A061348 .0.0.0
%e A061348 0.0.0.0
%p A061348 A061348 := proc(n) local t1, v, a; a := n*(n+1)/2; v := floor((n+1)/2); if n mod 3 = 1 then t1 := n*(n+1)/6+2/3; else t1 := n*(n+1)/6; fi; (1/6)*(2^a + 2*2^t1+3*2^(a/2+v/2)); end; # from Burnside's Lemma
%t A061348 A061348[n_] := Module[{t1, v, a}, a = n*(n+1)/2; v = Floor[(n+1)/2]; If[Mod[n, 3] == 1, t1 = n*(n+1)/6+2/3, t1 = n*(n+1)/6]; (1/6)*(2^a+2*2^t1+3*2^(a/2+v/2))]; Table[A061348[n], {n, 1, 15}] (* _Jean-François Alcover_, Feb 03 2014, after Maple *)
%Y A061348 Cf. A061709.
%K A061348 nonn,easy,nice
%O A061348 1,1
%A A061348 _Michel ten Voorde_, Jun 08 2001
%E A061348 Formula and more terms from _N. J. A. Sloane_, Jun 20 2001