|
%I #78 Nov 09 2022 08:21:15
%S 1,0,2,6,38,210,1314,8358,55118,370842,2541626,17668926,124321750,
%T 883614498,6334772562,45754956054,332639032734,2432189656362,
%U 17874332863722,131957567836206,978161729926950,7277592773408562,54327287358246018,406792963221032454
%N Number of negative formulas with two connectives (-> and *) and no variables.
%C More precisely, a(n) is the number of negative formulas containing n connectives * or ->, n+1 appearances of symbol "f"=false, and parentheses.
%C 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.
%C The total number of formulas, both affirmative and negative, with n connectives * or -> is A151374(n).
%H Vjekoslav Kovač, <a href="/A233130/b233130.txt">Table of n, a(n) for n = 0..1000</a>
%H V. Čačić and V. Kovač, <a href="http://arxiv.org/abs/1309.3408">On the fraction of IL formulas that have normal forms</a>, arXiv:1309.3408 [math.LO], 2013.
%F Recurrence: a(0)=1, a(n) = sum_{k=0..n-1} (A151374(k)-a(k)) a(n-k-1).
%F G.f.: (-3 - sqrt(1-8*x) + sqrt(10 + 56*x + 6*sqrt(1-8*x))) / (8*x).
%F The ratio a(n)/A151374(n) converges to 1/2 - 3*sqrt(17)/34 as n->infinity.
%F Asymptotics: a(n) ~ (1/2-3*sqrt(17)/34)*8^n/(sqrt(Pi)*n^(3/2)).
%e a(1)=0 because there are no negative formulas with 1 connective.
%e a(2)=2 because all negative formulas with 2 connectives are: (f->f)->f, (f*f)->f.
%e 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.
%t a[0] = 1;
%t For[n = 1, n <= 23, n++,
%t a[n] = Sum[(2^k Binomial[2 k, k]/(k + 1) - a[k]) a[n - k - 1], {k,
%t 0, n - 1}]];
%t Table[a[j], {j, 0, 23}]
%K nonn
%O 0,3
%A _Vjekoslav Kovac_, Dec 10 2013
|
