

A344900


a(n) is the number of wellformed formulas (wffs) of zerothorder 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
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OFFSET

1,3


COMMENTS

The characters of zerothorder logic include the tilde (~), ampersand (&), wedge (∨), horseshoe (⊃), triple bar (≡), left and right parentheses, and variables (uppercase letters with or without subscripts.) However, since the set of uppercase letters with or without subscripts is infinitely large, it is then, for the sentences of zerothorder 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..i1 (this gives a total of A000670(k1) legal permutations for a sentence containing k variables.)
The rules for a wellformed formula (wff) of zerothorder 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.


REFERENCES

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


LINKS



EXAMPLE

a(4) = 25, since the number of sentences of zerothorder 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.


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



