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%I #41 Aug 17 2021 10:47:38
%S 1,1,13,25,37,61,97,561,1869,4437,9097,17097,54101,194189,583857,
%T 1490017,3371997,8916485,28974361,94338361,277239461,728378813,
%U 1938657473,5839518033,18961970605,59883346869,174804016553,493085118121,1460284207861,4646560028141
%N a(n) is the number of well-formed formulas (wffs) of zeroth-order logic containing n characters (see comments).
%C The characters of zeroth-order logic include the tilde (~), ampersand (&), wedge (∨), horseshoe (⊃), triple bar (≡), left and right parentheses, and variables (upper-case letters with or without subscripts.) However, since the set of upper-case letters with or without subscripts is infinitely large, it is then, for the sentences of zeroth-order logic containing k variables, restricted to the set {A1, ..., Ak}, with an additional restriction as follows: a sentence may only contain Ai iff it contains every Aj for j=1..i-1 (this gives a total of A000670(k-1) legal permutations for a sentence containing k variables.)
%C The rules for a well-formed formula (wff) of zeroth-order logic are defined recursively as follows (see M. Bergmann et al.):
%C 1. Every variable is a wff.
%C 2. If P is a wff, then so is ~P.
%C 3. If P and Q are wffs, then so is (PxQ), where 'x' is any binary logical operator.
%C It is also customary to remove the outermost parentheses of a sentence.
%D Merrie Bergmann, James Moor, and Jack Nelson. The logic book. Vol. 2. New York: McGraw-Hill, 1990, p. 54.
%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a344/A344900.java">Java program</a> (github).
%e a(4) = 25, since the number of sentences of zeroth-order logic containing four characters are as follows: ~~~A, ~AxA, Ax~A, ~AxB, Bx~A, ~BxA, and Ax~B, where 'x' is any of the four binary logical operators.
%Y Cf. A300696, A000670.
%Y Related sequences: A101248, A101273, A308616. - _N. J. A. Sloane_, Aug 17 2021
%K nonn
%O 1,3
%A _Christoph B. Kassir_, Jun 01 2021
%E More terms from _Sean A. Irvine_, Jul 24 2021