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%I #10 Oct 01 2022 00:22:15
%S 1,1,2,3,4,4,5,6,6,7,8,9,9,10,11,12,13,13,13,14,15,16,17,18,19,19,19,
%T 19,20,21,22,23,24,25,26,27,28,28,28,28,28,29,30,31,32,33,34,35,36,37,
%U 38,39,40,41,41,41,41,41,41,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,60,60,60
%N a(n) = n - a^[3](n - a^[4](n-1)) with a(1) = 1, where a^[3](n) = a(a(a(n))) and a^[4](n) = a(a(a(a(n)))).
%C This is the third sequence in a family of nested-recurrent sequences with apparently similar structure defined as follows. Given a sequence s = {s(n): n >= 1} we define the k-th iterated sequence s^[k] by putting s^[1](n) = s(n) and setting s^[k](n) = s^[k-1](s(n)) for k >= 2. For k >= 1, we define a nested-recurrent sequence {u(n): n >= 1}, dependent on k, by putting u(1) = 1 and setting u(n) = n - u^[k](n - u^[k+1](n-1)) for n >= 2. The present sequence is the case k = 3. For other cases see A006165 (k = 1), A356988 (k = 2) and A356990 (k = 4).
%C The sequence is slow, that is, for n >= 1, a(n+1) - a(n) is either 0 or 1. The line graph of the sequence {a(n)} thus consists of a series of plateaus (where the value of the ordinate a(n) remains constant as n increases) joined by lines of slope 1.
%C The sequence of plateau heights begins 4, 6, 9, 13, 19, 28, 41, 60, ..., conjecturally A000930.
%C The plateaus start at absiccsa values n = 5, 8, 12, 17, 25, 37, 54, 79, ..., conjecturally A179070, and terminate at abscissa values n = 6, 9, 13, 19, 28, 41, 60, ..., conjecturally A000930.
%p a := proc (n) option remember; if n = 1 then 1 else n - a(a(a(n - a(a(a(a(n-1))))))) end if; end proc:
%p seq(a(n), n = 1..100);
%Y Cf. A000930, A006165, A179070, A356988, A356990.
%K nonn,easy
%O 1,3
%A _Peter Bala_, Sep 08 2022
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