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%I #37 Apr 03 2018 10:13:44
%S 1,2,8,46,350,3324,37874,503458,7648564,130722474,2482437926,
%T 51856030736,1181704007894,29172943488602,775597634145192,
%U 22093062633006326,671280598744505190,21671112459225274300,740767465663838556074,26727829360555847269034
%N a(n) is the number of n-place formulas in first-order logic when variables are allowed to coincide.
%C 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).
%C 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.
%C 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, ...
%H Tilman Piesk, <a href="/A300696/b300696.txt">Table of n, a(n) for n = 0..150</a>
%H Tilman Piesk, <a href="https://en.wikiversity.org/wiki/Formulas_in_predicate_logic/3_places">List of all 46 formulas with 3 places</a>
%H Tilman Piesk, <a href="https://github.com/watchduck/concertina_hypercubes/blob/master/cocoon.py">Python code used to generate the sequence</a>
%F a(0) = 1, a(n) = 2 * A083355(n) for n > 0.
%Y Cf. A000629, A083355, A300695.
%K nonn
%O 0,2
%A _Tilman Piesk_, Mar 13 2018
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