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.
2, 4, 20, 208, 5728, 351616, 44772352, 11453771776, 5864078802944, 6004800040206336, 12297829416834170880, 50371909152808594571264, 412646679762074900658913280, 6760803201217259503457555972096, 221537999297485988040673580072042496
Offset: 1
Examples
a(2) = 4, the labelings being {000}, {001}, {011}, {111}. Some of the 20 solutions for n=3 are as follows:
..0......1.......0......1.......1.......1.......0
.0.0....0.0.....1.0....1.0.....0.0.....0.0.....1.1
0.0.0..0.0.0...0.0.0..0.0.0...1.0.0...0.1.0...0.0.0
The first solution for n = 4 is
...0
..0.0
.0.0.0
0.0.0.0
Crossrefs
Cf. A061709.
Programs
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Maple
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
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Mathematica
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 *)
Formula
See Maple code for formula.
Extensions
Formula and more terms from N. J. A. Sloane, Jun 20 2001