cp's OEIS Frontend

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

%I A096365 #12 Mar 02 2025 16:03:59
%S A096365 0,2,3,4,5,5,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,
%T A096365 10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,
%U A096365 11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12
%N A096365 Maximum number of iterations of the RUNS transform needed to reduce any binary sequence of length n to a sequence of length 1.
%C A096365 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})
%e A096365 The following example shows that a(21)>=9:
%e A096365   x={100110100100110110100}
%e A096365   RUNS(x)={12211212212112}
%e A096365   RUNS^2(x)={1221121121}
%e A096365   RUNS^3(x)={1221211}
%e A096365   RUNS^4(x)={12112}
%e A096365   RUNS^5(x)={1121}
%e A096365   RUNS^6(x)={211}
%e A096365   RUNS^7(x)={12}
%e A096365   RUNS^8(x)={11}
%e A096365   RUNS^9(x)={2}
%e A096365 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.
%Y A096365 Cf. A319412.
%K A096365 nonn
%O A096365 1,2
%A A096365 _John W. Layman_, Jul 01 2004
%E A096365 More terms (using A319412 b-file) from _Pontus von Brömssen_, Mar 02 2025