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 A381324 #31 May 18 2025 22:24:24 %S A381324 3,17,177,3029,76713,2677637,122836857 %N A381324 Number of true implications over all possible pairs of unique logical sentences of n quantified variables in prenex normal form with a fixed proposition. %C A381324 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 A381324 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. %H A381324 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 A381324 Wikipedia, <a href="https://en.wikipedia.org/wiki/Prenex_normal_form">Prenex Normal Form</a> %F A381324 a(n) = A000629(n)^2 - A381325(n). %e A381324 a(1)=3, 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. %Y A381324 Cf. A000629, A381325. %K A381324 nonn,more %O A381324 1,1 %A A381324 _Adam Wang_, Feb 20 2025