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A276638 a(0)=0; thereafter a(n) is the number of matches between the first n terms circularly shifted by s and the reverse of the first n terms, maximized over s.

%I A276638 #22 Nov 08 2016 20:30:18
%S A276638 0,1,2,1,4,3,4,3,6,3,6,3,8,5,8,5,6,5,6,7,8,7,10,7,8,7,12,7,12,7,12,7,
%T A276638 12,7,14,9,14,9,14,9,14,9,14,9,14,11,16,11,14,11,16,11,18,11,16,11,16,
%U A276638 11,16,11,20,13,16,13,18,13,16,13,18,13,18,13,18
%N A276638 a(0)=0; thereafter a(n) is the number of matches between the first n terms circularly shifted by s and the reverse of the first n terms, maximized over s.
%C A276638 a(n) is the number of matches between (a(s), ..., a(n-1), a(0), ..., a(s-1)) and (a(n-1), ..., a(0)), maximized over s.
%C A276638 The odd bisection of the sequence (i. e., the subsequence a(2k+1)) appears to be bound both above and below by n^0.63 asymptotically. It includes odd terms only and grows monotonically with many plateaus.
%C A276638 The even bisection of the sequence (i. e., the subsequence a(2k)) appears to be bound both above and below asymptotically by the same power function as the odd bisection with larger coefficients. However, its behavior differs in other aspects: it includes even terms only and exhibits stochastic oscillations with increasing amplitude.
%H A276638 Andrey Zabolotskiy, <a href="/A276638/b276638.txt">Table of n, a(n) for n = 0..9999</a>
%e A276638 The first 6 terms (0, 1, 2, 1, 4, 3) shifted by 5 to the left yield (3, 0, 1, 2, 1, 4), which coincides with the first 6 terms reversed (3, 4, 1, 2, 1, 0) at 4 positions, and no shift produces more matches than 4, thus a(6)=4.
%t A276638 a = {0}; Do[AppendTo[a, Max@ Map[Count[Transpose@ #, w_ /; Equal @@ w] &, Array[{RotateLeft[a, #], Reverse@ a} &, n]]], {n, 72}]; a (* _Michael De Vlieger_, Sep 13 2016 *)
%o A276638 (Python)
%o A276638 a = [0]
%o A276638 for n in range(1, 100):
%o A276638     a.append(max(sum(a[(i+s)%n]==a[-i-1] for i in range(n)) for s in range(n)))
%Y A276638 Cf. A272727.
%K A276638 nonn,look
%O A276638 0,3
%A A276638 _Andrey Zabolotskiy_, Sep 08 2016