A339861 Lengths of runs of ones in A214323.

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, 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, 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

Offset

0,1

Comments

0 means two consecutive terms greater than 1 in A214323, so that every 'run' is separated by exactly one number greater than one.

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).

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.

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.

All of this applies regardless of the three starting terms used in A214551 or A000930, unless they all share a common divisor.

Links

Table of n, a(n) for n=0..128.

Example

The runs of ones in A214323 are:
(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, ...
Giving the terms:
6, 0, 0, 0, 0, 4, 0, 0, 0, 0, 6, 1, 4, 1, 1, 1, 0, 0, ...
Similarly the runs of consecutive numbers in A214653 are:
(0, 1, 2, 3, 4, 5), (11, 12, 13, 14), (20, 21, 22, 23, 24, 25), (27), (29, 30, 31, 32), (34), (36), (38), ...

Prog

(Python)
import math
a3 = a2 = a1 = 1
last_position = 0
run_lengths = []
for position in range(4, 20000):
    gcd = math.gcd(a1, a3)
    if gcd > 1:
        run_length = position - last_position - 1
        run_lengths.append(run_length)
        last_position = position
    a3, a2, a1 = a2, a1, (a1 + a3) // gcd
print(run_lengths)

Crossrefs

Cf. A214323, A214653, A214551, A214330.

Sequence in context: A228632 A229657 A306550 * A087936 A089804 A270606

Adjacent sequences: A339858 A339859 A339860 * A339862 A339863 A339864

Keyword

nonn

Author

Alex Hall, Apr 24 2021

Status

approved