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

0,2

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

Formula

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

Crossrefs

Cf. A000629, A083355, A300695.

Sequence in context: A052801 A294784 A180390 * A074599 A007289 A099765

Adjacent sequences: A300693 A300694 A300695 * A300697 A300698 A300699

Keyword

nonn

Author

Tilman Piesk, Mar 13 2018

Status

approved