Automatic-Sequence Proof of Cloitre's Self-Generating Sequence Density
Cloitre's self-generating sequence over {1,2} is defined by the rule that the sum of the terms in its n-th run equals twice its n-th term. This work shows the sequence is automatic -- computable by a finite automaton reading the base-2 representation of the index -- by connecting it to the regular paperfolding sequence, and uses that automaton to prove Cloitre's conjecture that the density of 1's in a length-n prefix is 2n/3 + o(n).
2501.00784
Cloitre's self-generating sequence over the alphabet {1,2} is the unique sequence, starting a1=1, in which the sum of the terms in the n-th maximal run equals twice the n-th term. This paper proves B…