A355316 Stuttering Look and Say sequence with seed 0.

0, 10, 1110, 333110, 333322110, 4444322222110, 444441355555222110, 5555541113555555333222110, 5555551433311366666653333333222110, 6666665111433332211366666661577777773333222110, 66666661533311444443222221137777777611157777777744443333222110

Offset: 1

Author

Jonathan Comes, Jun 28 2022

Keywords

• base
• nonn

Comments

If we let L(n) denote the number of digits in the n-th term, then the limit of L(n+1)/L(n) is an algebraic integer of degree 415. This limit is a stuttering analog of Conway's constant (see A014715).

Examples

			E.g., to obtain the term after 1110, we look at 1110 and see "three 1's and one 0". We then say what we saw by stuttering the counts as many times as the count prescribes: we stutter the "three" 3 times and the "one" 1 time (no stutter); so we say "three three three 1's and one 0" to get 333110.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..18
• Jonathan Comes, Stuttering look and say sequences and a challenger to Conway's most complicated algebraic number from the silliest source, arXiv:2206.11991 [math.HO], 2022.

Crossrefs

Stuttering variant of A001155.

Cf. A014715.

Programs

PARI

first(n) = my(c, d=[0], x, res=vector(n)); for(i=2, n, c=1; x=""; for(j=1, #d, if(j<#d && d[j]==d[j+1], c++, x=concat(x, concat(vector(c+1, k, Str(if(k==c+1, d[j], c))))); c=1)); res[i]=eval(x); d=digits(res[i])); res \\ Iain Fox, Jun 30 2022

Python

from itertools import accumulate, groupby, repeat
def summarize(n, _): return int("".join(str(c:=len(list(g)))*c+k for k, g in groupby(str(n))))
def aupton(terms): return list(accumulate(repeat(0, terms), summarize))
print(aupton(11)) # Michael S. Branicky, Jun 28 2022