A344900 a(n) is the number of well-formed formulas (wffs) of zeroth-order logic containing n characters (see comments).

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

Offset: 1

Author

Christoph B. Kassir, Jun 01 2021

Keywords

• nonn

Comments

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.

Examples

			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.
		

References

• Merrie Bergmann, James Moor, and Jack Nelson. The logic book. Vol. 2. New York: McGraw-Hill, 1990, p. 54.

Links

• Sean A. Irvine, Java program (github).

Crossrefs

Cf. A300696, A000670.

Related sequences: A101248, A101273, A308616. - N. J. A. Sloane, Aug 17 2021

Extensions

More terms from Sean A. Irvine, Jul 24 2021