A356990 - OEIS

%I #9 Oct 01 2022 13:11:25

%S 1,1,2,3,4,5,5,6,7,7,8,9,10,10,11,12,13,14,14,15,16,17,18,19,19,19,20,

%T 21,22,23,24,25,26,26,26,26,27,28,29,30,31,32,33,34,35,36,36,36,36,36,

%U 37,38,39,40,41,42,43,44,45,46,47,48,49,50,50,50,50,50,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,69,69,69,69,69,69,69,70,71

%N a(n) = n - a^[4](n - a^[5](n-1)) with a(1) = 1, where a^[4](n) = a(a(a(a(n)))) and a^[5](n) = a(a(a(a(a(n))))).

%C This is the fourth 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](u(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. This is the case k = 4. For other cases see A006165 (k = 1), A356988 (k = 2) and A356989 (k = 3).

%C The sequence is slow, that is, for n >= 1, a(n+1) - a(n) is either 0 or 1.

%C 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 5, 7, 10, 14, 19, 26, 36, 50, ..., which appears to be A003269.

%C The plateaus start at absiccsa values n = 6, 9, 13, 18, 24, 33, 46, 64, ..., which appears to be A014101, and terminate at abscissa values 7, 10, 14, 19, 26, 36, 50, ..., conjecturally A003269.

%p a := proc(n) option remember; if n = 1 then 1 else n - a(a(a(a(n - a(a(a(a(a(n-1))))))))) end if; end proc:

%p seq(a(n), n = 1..100);

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A356990(n): return 1 if n <= 1 else n-A356990(A356990(A356990(A356990(n-A356990(A356990(A356990(A356990(A356990(n-1))))))))) # _Chai Wah Wu_, Oct 01 2022

%Y Cf. A003269, A006165, A014101, A356988, A356989.

%K nonn,easy

%O 1,3

%A _Peter Bala_, Sep 08 2022