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A344900 a(n) is the number of well-formed formulas (wffs) of zeroth-order logic containing n characters (see comments). 1
1, 1, 13, 25, 37, 61, 97, 561, 1869, 4437, 9097, 17097, 54101, 194189, 583857, 1490017, 3371997, 8916485, 28974361, 94338361, 277239461, 728378813, 1938657473, 5839518033, 18961970605, 59883346869, 174804016553, 493085118121, 1460284207861, 4646560028141 (list; graph; refs; listen; history; text; internal format)
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.)
The rules for a well-formed formula (wff) of zeroth-order logic are defined recursively as follows (see M. Bergmann et al.):
1. Every variable is a wff.
2. If P is a wff, then so is ~P.
3. If P and Q are wffs, then so is (PxQ), where 'x' is any binary logical operator.
It is also customary to remove the outermost parentheses of a sentence.
Merrie Bergmann, James Moor, and Jack Nelson. The logic book. Vol. 2. New York: McGraw-Hill, 1990, p. 54.
Sean A. Irvine, Java program (github).
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.
Related sequences: A101248, A101273, A308616. - N. J. A. Sloane, Aug 17 2021
Sequence in context: A309809 A264732 A298483 * A195558 A026058 A032697
Christoph B. Kassir, Jun 01 2021
More terms from Sean A. Irvine, Jul 24 2021

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Last modified June 4 02:25 EDT 2023. Contains 363118 sequences. (Running on oeis4.)