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

0,3

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.

Links

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

Example

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.

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 *)

Prog

(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)))

Crossrefs

Cf. A272727.

Sequence in context: A130973 A093779 A231898 * A116449 A316433 A071046

Adjacent sequences: A276635 A276636 A276637 * A276639 A276640 A276641

Keyword

nonn,look

Author

Andrey Zabolotskiy, Sep 08 2016

Status

approved