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Sum, over all binary strings w of length n, of the length of the shortest border of w.
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%I #16 Jan 16 2020 01:57:52

%S 0,2,4,12,24,60,120,264,528,1116,2232,4584,9168,18616,37232,75056,

%T 150112,301556,603112,1209064,2418128,4842504,9685008,19383408,

%U 38766816,77562648,155125296,310312528,620625056,1241382832,2482765664,4965813280,9931626560

%N Sum, over all binary strings w of length n, of the length of the shortest border of w.

%C 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 From _Rémy Sigrist_, Jan 16 2020: (Start)

%F Apparently, for any k > 0:

%F - a(2*k+1) = 2*a(k),

%F - a(2*k) = 2*a(2*k-1) + 2*k*A045690(k).

%F (End)

%e 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 Cf. A045690, A091065, A331393.

%K nonn

%O 1,2

%A _Jeffrey Shallit_, Jan 15 2020

%E More terms from _Rémy Sigrist_, Jan 15 2020