A300696 a(n) is the number of n-place formulas in first-order logic when variables are allowed to coincide.

1, 2, 8, 46, 350, 3324, 37874, 503458, 7648564, 130722474, 2482437926, 51856030736, 1181704007894, 29172943488602, 775597634145192, 22093062633006326, 671280598744505190, 21671112459225274300, 740767465663838556074, 26727829360555847269034

Offset: 0

Author

Tilman Piesk, Mar 13 2018

Keywords

• nonn

Comments

An example of a 3-place formula in predicate logic is Ex Ay Ez P(x,y,z). The number of different formulas when x, y, z have to be different is A000629(3) = 26. When variables are allowed to coincide that means that there are 20 more formulas like, e.g., Ex Ay P(x,x,y) or Ex P(x,x,x).

a(n) is the number of vertices in a cocoon concertina n-cube and the sum of row n in A300695, which shows the number of vertices in that structure by rank. A000629(n) by comparison is the number of vertices in the convex concertina n-cube.

The differences with A000629, i.e., the numbers of formulas with coinciding variables, are 0, 0, 2, 20, 200, 2242, 28508, 408872, 6556894, 116547952, 2277942800, ...

Links

• Tilman Piesk, Table of n, a(n) for n = 0..150
• Tilman Piesk, List of all 46 formulas with 3 places
• Tilman Piesk, Python code used to generate the sequence

Crossrefs

Cf. A000629, A083355, A300695.

Formula

a(0) = 1, a(n) = 2 * A083355(n) for n > 0.