cp's OEIS Frontend

A306559 A self-describing sequence mostly made of 1's and 0's emerging when written in English words (see the Comments section for an explanation).

%I A306559 #39 Apr 06 2020 00:14:30
%S A306559 1,2,3,0,1,1,0,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,
%T A306559 1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,
%U A306559 0,1,1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0
%N A306559 A self-describing sequence mostly made of 1's and 0's emerging when written in English words (see the Comments section for an explanation).
%C A306559 We start with 1, 2, 3. From this point on, the sequence is extended by the rule that when written in English words, the lengths of runs of consonants between successive vowels is the sequence itself. The sequence is ONE, TWO, THREE, ZERO, ONE, ONE, ZERO, ONE, ZERO, ONE, ONE, ONE, ZERO, ONE, ONE, ONE, ZERO, ...
%C A306559 Those runs of consonants are better viewed with commas and spaces deleted: ONETWOTHREEZEROONEONEZEROONEZEROONEONEONEZEROONEONEONEZERO...
%C A306559 Note that there is no (thus ZERO) consonant between the two E's of THREE.
%C A306559 Jean-Marc Falcoz has computed 100000 terms and found no place where the sequence enters into a loop. It is conjectured that the sequence will grow infinitely without ever entering into such a loop.
%C A306559 The proportions of ONEs and ZEROs are conjectured to be 2/3 and 1/3.
%C A306559 _Alan C. Wechsler_ (Feb 23 2019) observes that if the first four terms are deleted, this appears to coincide with A039982. Indeed, the resulting pair of sequences agree for at least the first 28442 terms, so this is a very plausible conjecture. - _N. J. A. Sloane_, Feb 23 2019
%C A306559 After deleting the first three terms, this sequence is 0*W(0)*W^2(0)*W^3(0)*... where W is the morphism 0 -> 11, 1 -> 01. It can be shown that this is equivalent to the above conjecture (see link for proof). - _Charlie Neder_, Mar 04 2019
%H A306559 Jean-Marc Falcoz, <a href="/A306559/b306559.txt">Table of n, a(n) for n = 1..28446</a>
%H A306559 Charlie Neder, <a href="/A306559/a306559.txt">Proof of the equivalence of this sequence and A039982</a>
%e A306559 The #1 vowel (O) and #2 vowel (E) enclose indeed ONE consonant (N);
%e A306559 the #2 vowel (E) and #3 vowel (O) enclose indeed TWO consonants (TW);
%e A306559 the #3 vowel (O) and #4 vowel (E) enclose indeed THREE consonants (THR);
%e A306559 the #4 vowel (E) and #5 vowel (E) enclose indeed ZERO consonant;
%e A306559 the #5 vowel (E) and #6 vowel (E) enclose indeed ONE consonant (Z);
%e A306559 the #6 vowel (E) and #7 vowel (O) enclose indeed ONE consonant (R);
%e A306559 the #7 vowel (O) and #8 vowel (O) enclose indeed ZERO consonant;
%e A306559 etc.
%Y A306559 Cf. A039982.
%K A306559 base,nonn,word
%O A306559 1,2
%A A306559 _Eric Angelini_ and _Jean-Marc Falcoz_, Feb 23 2019