A300695 -id:A300695 - cp's OEIS frontend

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

Tilman Piesk, Mar 13 2018

nonn

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, ...

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

Cf. A000629, A083355, A300695.

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

A300699 Irregular triangle read by rows: T(n, k) = number of vertices with rank k in concertina n-cube.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 6, 6, 3, 1, 1, 4, 12, 18, 28, 24, 28, 18, 12, 4, 1, 1, 5, 20, 40, 80, 95, 150, 150, 150, 150, 95, 80, 40, 20, 5, 1, 1, 6, 30, 75, 180, 270, 506, 660, 840, 1080, 1035, 1035, 1080, 840, 660, 506, 270, 180, 75, 30, 6, 1, 1, 7, 42, 126, 350, 630, 1337, 2107, 3192, 4760

Offset: 0

Tilman Piesk, Mar 11 2018

n-place formulas in first-order logic like Ax Ey P(x, y) ordered by implication form a graded poset, and its Hasse diagram is the concertina n-cube.
Sum of row n is A000629(n), the number of vertices of a concertina n-cube.
The rows are palindromic. Their lengths are the central polygonal numbers A000124 = 1, 2, 4, 7, 11, 16, 22, ... That means after row 0 rows of even and odd length follow each other in pairs.
The central values are 1, (1), (2), 6, 24, (150), (1035), 9030, 88760, (1002204), ... (Values next to the center in rows of even length are in parentheses.)
Maximal values are 1, 1, 2, 6, 28, 150, 1080, 9030, 88760, 1002204, ...
A300695 is a triangle of the same shape that shows the number of ranks in cocoon concertina hypercubes.

			First rows of the triangle:
    k   0   1   2   3   4   5    6    7    8    9   10  11  12  13  14  15
  n
  0     1
  1     1   1
  2     1   2   2   1
  3     1   3   6   6   6   3    1
  4     1   4  12  18  28  24   28   18   12    4    1
  5     1   5  20  40  80  95  150  150  150  150   95  80  40  20   5   1
  6     1   6  30  75 180 270  506  660  840 1080 1035 ...
The ranks of vertices of a concertina cube (n=3) can be seen in the linked Hasse diagrams. T(3, 4) = 6, so there are 6 vertices with rank 4.
Ey Ez Ax P(x, y, z) implies Ey Ax Ez P(x, y, z), and their ranks are 3 and 4. As the difference in rank is 1, this implication is an edge in the Hasse diagram.

Tilman Piesk, Rows 0..9, flattened
Tilman Piesk, Rows 0..9
Tilman Piesk, Formulas in predicate logic (Wikiversity)
Tilman Piesk, Concertina cube Hasse diagram with labels and with highlighted ranks
Tilman Piesk, Lists of vertices ordered by rank for n=2..6
Tilman Piesk, Python code used to generate the sequence

Cf. A000124, A000629, A300695.

cp's OEIS Frontend

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

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