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 A331392 #16 Jan 16 2020 01:57:52 %S A331392 0,2,4,12,24,60,120,264,528,1116,2232,4584,9168,18616,37232,75056, %T A331392 150112,301556,603112,1209064,2418128,4842504,9685008,19383408, %U A331392 38766816,77562648,155125296,310312528,620625056,1241382832,2482765664,4965813280,9931626560 %N A331392 Sum, over all binary strings w of length n, of the length of the shortest border of w. %C A331392 A nonempty word w is a border of a string x if w is both a prefix and suffix of x, and w does not equal x. %F A331392 From _Rémy Sigrist_, Jan 16 2020: (Start) %F A331392 Apparently, for any k > 0: %F A331392 - a(2*k+1) = 2*a(k), %F A331392 - a(2*k) = 2*a(2*k-1) + 2*k*A045690(k). %F A331392 (End) %e A331392 For n = 3, the words are 000,001,010,011 and their binary complements. The shortest border of 000 and 010 is 0, and the other words have no border. So a(3) = 4. %Y A331392 Cf. A045690, A091065, A331393. %K A331392 nonn %O A331392 1,2 %A A331392 _Jeffrey Shallit_, Jan 15 2020 %E A331392 More terms from _Rémy Sigrist_, Jan 15 2020