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A339861 Lengths of runs of ones in A214323.

%I A339861 #9 Apr 24 2021 21:54:33
%S A339861 6,0,0,0,0,4,0,0,0,0,6,1,4,1,1,1,0,0,0,4,0,3,0,1,0,0,0,0,0,0,3,2,3,2,
%T A339861 6,1,6,1,3,0,1,1,1,1,6,1,6,0,0,1,1,1,1,0,0,2,2,0,1,1,1,1,6,5,0,5,0,5,
%U A339861 0,1,1,1,0,0,5,0,1,1,2,3,1,1,1,0,1,4,1,1,1,1,1,1,0,1,1,1,1,6,1,1,1,1,2,1,6,1,5,0,0,1,1,1,6,1,1,1,1,1,1,1,6,1,4,0,1,3,1,1,1
%N A339861 Lengths of runs of ones in A214323.
%C A339861 0 means two consecutive terms greater than 1 in A214323, so that every 'run' is separated by exactly one number greater than one.
%C A339861 Equivalently, this gives the lengths of runs of consecutive numbers in A214653 (ignoring the zeros in this sequence), indicating consecutive coprime pairs (A214551(n-1), A214551(n-2)), which lead to A214551 increasing (although it can also increase after a non-coprime pair).
%C A339861 Empirically, the most common term is 1, then 0, then 6, but provably there is no term higher than 6. This can be understood by looking at A214330 and the state diagram. The state 010 (a pair of even numbers in A214551) is always separated by exactly 1 or 6 other states, i.e., even divisors in A214323 are always separated by exactly 1 or 6 odd divisors.
%C A339861 If instead you consider runs of ones in gcd( A000930(n-1), A000930(n-3) ) (i.e., don't divide by the gcd but still observe it) then the maximum run length of ones is still provably 6, but empirically longer runs appear consistently less often than shorter runs as you'd expect.
%C A339861 All of this applies regardless of the three starting terms used in A214551 or A000930, unless they all share a common divisor.
%e A339861 The runs of ones in A214323 are:
%e A339861 (1, 1, 1, 1, 1, 1), 2,() 3,() 2,() 3,() 2, (1, 1, 1, 1), 7,() 3,() 2,() 3,() 4, (1, 1, 1, 1, 1, 1), 4, (1), 5, (1, 1, 1, 1), 8, (1), 2, (1), 6, (1), 4,() 5,() 4, ...
%e A339861 Giving the terms:
%e A339861 6, 0, 0, 0, 0, 4, 0, 0, 0, 0, 6, 1, 4, 1, 1, 1, 0, 0, ...
%e A339861 Similarly the runs of consecutive numbers in A214653 are:
%e A339861 (0, 1, 2, 3, 4, 5), (11, 12, 13, 14), (20, 21, 22, 23, 24, 25), (27), (29, 30, 31, 32), (34), (36), (38), ...
%o A339861 (Python)
%o A339861 import math
%o A339861 a3 = a2 = a1 = 1
%o A339861 last_position = 0
%o A339861 run_lengths = []
%o A339861 for position in range(4, 20000):
%o A339861     gcd = math.gcd(a1, a3)
%o A339861     if gcd > 1:
%o A339861         run_length = position - last_position - 1
%o A339861         run_lengths.append(run_length)
%o A339861         last_position = position
%o A339861     a3, a2, a1 = a2, a1, (a1 + a3) // gcd
%o A339861 print(run_lengths)
%Y A339861 Cf. A214323, A214653, A214551, A214330.
%K A339861 nonn
%O A339861 0,1
%A A339861 _Alex Hall_, Apr 24 2021