This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381325 #28 May 18 2025 22:22:38 %S A381325 1,19,499,19471,1094011,85044319,8823674539 %N A381325 Number of false implications over all possible pairs of unique logical sentences of n quantified variables in prenex normal form with a fixed proposition. %C A381325 The total number of unique logical sentences of n quantified variables in prenex normal form (PNF) with a fixed proposition is given by A000629. Essentially, a logical sentence is in PNF iff it is a string of quantifiers followed by a proposition. %C A381325 Note that for an arbitrary proposition, the only two possible implications are: firstly, "for all x_1" -> "exists x_1", and, secondly, "exists x_1 forall x_2" -> "forall x_2 exists x_1". The sequence is formed by counting all the number of implications between all valid PNFs for a fixed proposition. %C A381325 For example, a(1)=1, because "forall x P(x)" and "exists x P(x)" both imply themselves, and the former implies the latter. However, the latter does not imply the former. %H A381325 Adam Wang, <a href="https://arxiv.org/abs/2504.15294">Determining Implication of Fixed Matrix Prenex Normal Forms Can Be Decided in Linear Time</a>, arXiv:2504.15294 [cs.DS], 2025. %H A381325 Wikipedia, <a href="https://en.wikipedia.org/wiki/Prenex_normal_form">Prenex Normal Form</a> %F A381325 a(n) = A000629(n)^2 - A381324(n). %Y A381325 Cf. A000629, A381324. %K A381325 nonn,more %O A381325 1,2 %A A381325 _Adam Wang_, Feb 20 2025