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 A243841 #21 Mar 23 2024 17:30:53 %S A243841 0,0,1,0,0,0,0,0,-1,0,0,1,1,0,0,0,1,0,0,0,0,0,-1,0,0,1,1,1,0,1,1,0,0, %T A243841 0,1,0,0,0,1,1,2,1,0,1,1,1,0,1,1,1,0,1,1,1 %N A243841 Pair deficit of the most nearly equal partition of n into two parts using ceiling rounding of the expectations of n, floor(n/2) and n-floor(n/2), assuming equal likelihood of states defined by the number of 2-cycles. %C A243841 A162970 and A000085 provide the numerator and the denominator for calculating the expected value. %F A243841 a(n) = ceiling(A162970(n)/A000085(n)) - (ceiling(A162970(floor(n/2))/A000085(floor(n/2))) + ceiling(A162970(n-floor(n/2))/A000085(n-floor(n/2)))). %e A243841 Trivially, for n = 0,1 no pairs are possible so a(0) and a(1) are 0. %e A243841 For n = 2, the expectation, E(n), equals 0.5. So a(2) = ceiling(E(2)) - (ceiling(E(1)) + ceiling(E(1))) = 1. %e A243841 For n = 5 = 2 + 3, E(5) = 20/13, E(2) = 0.5 and E(3) = 0.75 and a(5) = ceiling(E(5)) - (ceiling(E(2)) + ceiling(E(3))) = 0. %e A243841 Interestingly, for n = 8, E(8) = 532/191 and E(4) = 6/5, so a(n) = 3 - (2 + 2) = -1. %Y A243841 Cf. A000085, A162970. %K A243841 sign,more %O A243841 0,41 %A A243841 _Rajan Murthy_, Jun 12 2014