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 A384561 #28 Jun 11 2025 00:32:32 %S A384561 1,6,39,284,2337,21474,218179,2430216,29459301,386182478,5444570631, %T A384561 82157021556,1321282006249,22562446559034,407722012334667, %U A384561 7773697259015264,155956589714240109,3284208113313605286,72434065593967762831,1669777527837108720588,40157785493048522566641 %N A384561 One fourth of the number of permutations of [n] with |p(i+1) - p(i)| >= 2, for i = 1..(n-1) and n appears at position i = 1 or i = n. %C A384561 The number of such permutations of [n] is 1 for n = 1 (the p(i) condition is not needed), and 0 for n = 2, 3 and 4, hence a(1) = 1/4 and a(n) = 0 for n = 2, 3 and 4. %C A384561 The number of permutations of [n] with |p(i+1) - p(i)| >= 2, for i = 1..(n-1), for n >= n is given by A002464(n), for n >= 0. See also A001266(n) = A002464(n)/2, for n >= 2. These permutations are also called king permutations, e.g., in A382644. %F A384561 a(n) = A382644(n-1)/2, for n >= 5. %F A384561 a(n) = (A001266(n) - A242522(n+1))/2, for n >= 5. %F A384561 a(n) = A382644(n)/2 - A242522(n+1), for n >= 5. %F A384561 a(n) = a(n-1) + A242522(n), for n >= 6, with a(5) = 1. %e A384561 n = 5: the 4 permutations are 2 4 1 3 5, 3 1 4 2 5 and their reversals 5 3 1 4 2, 5 2 4 1 3. %e A384561 a(5) = 1 = A382644(4)/2 = (A001236(5) - A242522(6))/2 = (7 - 5)/2, and A382644(5)/2 - A242522(6) = 6 - 5 = 1 %e A384561 a(6) = a(5) + A242522(6) = 1 + 5 = 6. %Y A384561 Cf. A002464, A001266, A242522, A382644. %K A384561 nonn,easy %O A384561 5,2 %A A384561 _Wolfdieter Lang_, Jun 04 2025