A096365 Maximum number of iterations of the RUNS transform needed to reduce any binary sequence of length n to a sequence of length 1.

0, 2, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9

Offset

1,2

Comments

The RUNS transform maps a finite word (or sequence) x to the (finite) sequence y whose i-th term is the length of the i-th subsequence of consecutive identical terms of x. (Example: RUNS{1,2,2,2,1,1,3,3,1}={1,3,2,2,1})

Links

Table of n, a(n) for n=1..21.

Example

The following example shows that a(21)>=9:
x={100110100100110110100}
RUNS(x)={12211212212112}
RUNS^2(x)={1221121121}
RUNS^3(x)={1221211}
RUNS^4(x)={12112}
RUNS^5(x)={1121}
RUNS^6(x)={211}
RUNS^7(x)={12}
RUNS^8(x)={11}
RUNS^9(x)={2}
Since calculation shows that no other binary sequence of length 21 requires more than 9 iterations of RUNS to reduce it to a single term, we have a(21)=9.

Crossrefs

Cf. A319412

Sequence in context: A306560 A025280 A365092 * A319412 A200311 A007600

Adjacent sequences: A096362 A096363 A096364 * A096366 A096367 A096368

Keyword

nonn

Author

John W. Layman, Jul 01 2004

Status

approved