OFFSET
0,3
COMMENTS
More precisely, a(n) is the number of negative formulas containing n connectives * or ->, n+1 appearances of symbol "f"=false, and parentheses.
Each negative formula is either "f", or is of the form "A->N", where A is a simpler affirmative formula and N is a simpler negative formula. Affirmative formulas are precisely those that are not negative.
The total number of formulas, both affirmative and negative, with n connectives * or -> is A151374(n).
LINKS
Vjekoslav Kovač, Table of n, a(n) for n = 0..1000
V. Čačić and V. Kovač, On the fraction of IL formulas that have normal forms, arXiv:1309.3408 [math.LO], 2013.
FORMULA
EXAMPLE
a(1)=0 because there are no negative formulas with 1 connective.
a(2)=2 because all negative formulas with 2 connectives are: (f->f)->f, (f*f)->f.
a(3)=6 because all negative formulas with 3 connectives are: ((f->f)*f)->f, ((f*f)*f)->f, (f->(f->f))->f, (f->(f*f))->f, (f*(f->f))->f, (f*(f*f))->f.
MATHEMATICA
a[0] = 1;
For[n = 1, n <= 23, n++,
a[n] = Sum[(2^k Binomial[2 k, k]/(k + 1) - a[k]) a[n - k - 1], {k,
0, n - 1}]];
Table[a[j], {j, 0, 23}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vjekoslav Kovac, Dec 10 2013
STATUS
approved