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.

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, 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, 11, 16, 11, 20, 13, 16, 13, 18, 13, 16, 13, 18, 13, 18, 13, 18

Offset: 0

Author

Andrey Zabolotskiy, Sep 08 2016

Keywords

• nonn
• look

Comments

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.

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.

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.

Examples

			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.
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 0..9999

Crossrefs

Cf. A272727.

Programs

Mathematica

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 *)

Python

a = [0]
for n in range(1, 100):
    a.append(max(sum(a[(i+s)%n]==a[-i-1] for i in range(n)) for s in range(n)))