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A134548 a(1)=1, a(n) = 2 + maximal digit of Sum_{j=1..n-1} a(j).

%I A134548 #11 May 19 2025 12:02:29
%S A134548 1,3,6,3,5,10,10,10,10,10,10,10,10,11,11,4,6,5,7,6,10,10,10,10,10,11,
%T A134548 11,4,6,5,7,6,10,10,10,10,10,11,11,5,7,5,9,8,7,8,11,10,11,6,9,8,6,6,8,
%U A134548 6,7,9,8,9,10,11,7,7,10,10,10,10,10,10,10,10,11,11,8,10,10,10,10,10,10,10
%N A134548 a(1)=1, a(n) = 2 + maximal digit of Sum_{j=1..n-1} a(j).
%C A134548 For n > 1, 3 <= a(n) <= 11 by definition. Also subsequences with repeating terms may be arbitrarily long. What about any other patterns, periods, etc.?
%H A134548 Robert Israel, <a href="/A134548/b134548.txt">Table of n, a(n) for n = 1..10000</a>
%H A134548 Robert Israel, <a href="/A134548/a134548.png">Plot of a(n) for n = 1 .. 5*10^5</a>
%p A134548 s:= proc(n) option remember;
%p A134548    procname(n-1) + a(n-1)
%p A134548 end proc:
%p A134548 s(1):= 0:
%p A134548 a:=  proc(n) 2 + max(convert(s(n),base,10)) end proc:
%p A134548 a(1):= 1:
%p A134548 map(a, [$1..100]); # _Robert Israel_, May 19 2025
%t A134548 a[1]=i=s=1;Do[i++;b=2+Max[IntegerDigits[s]];a[i]=b;s+=b,{145}]; Table[a[k],{k,i}]
%K A134548 base,nonn,look
%O A134548 1,2
%A A134548 _Zak Seidov_, Oct 31 2007