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A356992 Then a(n) = n - b(n - b(n - b(n - b(n - b(n - b(n)))))) for n >= 2, where b(n) = A356988(n).

%I A356992 #10 Oct 13 2022 12:53:36
%S A356992 1,2,3,4,4,4,5,6,7,7,7,8,9,10,11,11,11,11,12,13,14,15,16,17,18,18,18,
%T A356992 18,18,18,19,20,21,22,23,24,25,26,27,28,29,29,29,29,29,29,29,29,29,30,
%U A356992 31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,47,47,47,47,47,47,47,47,47,47,47,47,47,48
%N A356992 Then a(n) = n - b(n - b(n - b(n - b(n - b(n - b(n)))))) for n >= 2, where b(n) = A356988(n).
%C A356992 The sequence is slow, that is, for n >= 2, a(n+1) - a(n) is either 0 or 1. The sequence is unbounded.
%C A356992 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 A356992 The sequence of plateau heights begins 4, 7, 11, 18, 29, 47, ..., conjecturally the Lucas sequence {A000032(k): k >= 3}.
%C A356992 The plateaus start at abscissa values n = 5, 10, 16, 26, 42, 68, .... Apart from the first term 5, this appears to be the sequence {2*Fibonacci(k): k >= 5}.
%C A356992 The plateaus end at abscissa values n = 7, 12, 19, 31, 50, 81, ..., conjecturally the sequence {A013655(k): k >= 3}.
%C A356992 The sequence of plateau lengths begins 2, 2, 3, 5, 8, 13, .... Apart from the first term 2, this appears to be the sequence {Fibonacci(k): k >= 3}.
%C A356992 The slow sequences {a(a(n)): n >= 3} and {a(a(a(n))): n >= 4} appear to have similar properties to the present sequence. The slow sequence {n - a(n): n >= 2} appears to have plateaus at heights given by the Fibonacci sequence. See the Example section.
%e A356992 Related sequences:
%e A356992 1) {n - a(n): n >= 2}
%e A356992   1, 1, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 35, 36, 37, ...
%e A356992 The line graph of the sequence has plateaus at heights 3, 5, 8, 13, 21, 34, ..., conjecturally the Fibonacci numbers A000045.
%e A356992 2) {a(a(n)): n >= 3}
%e A356992   1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 8, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 47, 47, 47, 47, ...
%e A356992 The line graph of the sequence has plateaus at heights 3, 4, 7, 11, 28, 29, ..., conjecturally the Lucas numbers A000045.
%e A356992 3) {a(a(a(n))): n >= 4}
%e A356992   1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 29, 29, ...
%e A356992 The line graph of the sequence has plateaus at heights (2), 3, 4, 7, 11, 28, 29, ..., conjecturally the Lucas numbers A000045.
%p A356992 b := proc(n) option remember; if n = 1 then 1 else n - b(b(n - b(b(b(n-1))))) end if; end proc:
%p A356992 seq(n - b(n - b(n - b(n - b(n - b(n - b(n)))))), n = 2..100);
%Y A356992 Cf. A000032, A000045, A013655, A356988, A356991 - A356999.
%K A356992 nonn,easy
%O A356992 2,2
%A A356992 _Peter Bala_, Sep 08 2022