A309318
a(n) is the number of polygons whose vertices are the (2*n+1)-th roots of unity and whose 2*n+1 sides all have distinct slopes.
Original entry on oeis.org
1, 2, 24, 180, 2700, 74184, 2062800, 81067840, 3912595776
Offset: 1
For n=2, the a(2)=2 solutions for 2*2+1 = 5 sides are the regular pentagon and pentagram.
A340234
Number of harmonious graphs with n edges and at most n vertices, allowing self-loops.
Original entry on oeis.org
1, 2, 8, 36, 243, 1728, 16384, 160000, 1953125, 24300000, 362797056, 5489031744, 96889010407, 1727094849536, 35184372088832, 722204136308736, 16677181699666569, 387420489000000000, 10000000000000000000, 259374246010000000000
Offset: 1
For n=3, the a(3) = 8 solutions are represented by the following adjacency matrices:
0 1 2 0 1 2 0 1 2 0 1 2
0 [ 1 1 1 ] 0 [ 1 1 0 ] 0 [ 1 0 1 ] 0 [ 1 0 0 ]
1 [ 1 0 0 ] 1 [ 1 1 0 ] 1 [ 0 0 0 ] 1 [ 0 1 0 ]
2 [ 1 0 0 ] 2 [ 0 0 0 ] 2 [ 1 0 1 ] 2 [ 0 0 1 ]
0 1 2 0 1 2 0 1 2 0 1 2
0 [ 0 1 1 ] 0 [ 0 1 0 ] 0 [ 0 0 1 ] 0 [ 0 0 0 ]
1 [ 1 0 1 ] 1 [ 1 1 1 ] 1 [ 0 0 1 ] 1 [ 0 1 1 ]
2 [ 1 1 0 ] 2 [ 0 1 0 ] 2 [ 1 1 1 ] 2 [ 0 1 1 ]
Notice that the number of self-loops in each graph is equal to the sum of the main diagonal.
For n odd,
A110654 to the n-th power.
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