cp's OEIS Frontend

A356991 a(n) = b(n) + b(n - b(n)) for n >= 2, where b(n) = A356998(n).

%I A356991 #14 Oct 13 2022 12:53:26
%S A356991 2,3,4,4,5,6,7,8,9,10,11,11,12,13,14,15,16,17,18,18,19,20,21,22,23,24,
%T A356991 25,26,27,28,29,29,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,
%U A356991 46,47,47,47,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,76,76,76,76,76,77,78,79,80
%N A356991 a(n) = b(n) + b(n - b(n)) for n >= 2, where b(n) = A356998(n).
%C A356991 The sequence is slow, that is, for n >= 2, a(n+1) - a(n) is either 0 or 1. The sequence is unbounded.
%C A356991 The line graph of the sequence {a(n)} thus consists of a series of plateaus (where the value of the ordinate a(n) is unchanged with increasing values of the abscissa n) joined by lines of slope 1.
%C A356991 The sequence of plateau heights begins 4, 7, 11, 18, 29, 47, 76, 123, 199, ..., the Lucas sequence {A000032(k): k >= 3}. The plateaus start at absiccsa values n = 4, 8, 12, 20, 32, 52, 84, 136, ..., the sequence {A022087(k): k >= 2}, and end at abscissa values n = 5, 8, 13, 21, 34, 55, 89, ..., the Fibonacci sequence {A000045(k): k >= 5}.
%C A356991 Compare with A356992 and A356993.
%C A356991 Other sequences defined in terms of b(n) = A356998(n) that are similarly related to the Lucas numbers include {n - b(b(b(2*n - b(n)))): n >= 1} beginning 0, 1, 2, 3, 3, 4, 4, 5, 6, 7, 7, 7, 8, 9, 10, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 17, 18, 19, ... and {2*n - b(2*n - b(2*n - b(n))) : n >= 1} beginning 1, 3, 4, 5, 7, 7, 9, 11, 11, 12, 14, 16, 18, 18, 18, 19, 21, 23, 25, 27, 29, 29, 29, 29, 29, 31, .... Neither sequence is slow.
%F A356991 The sequence is completely determined by the initial values a(2) = 2, a(3) = 3 and the pair of formulas:
%F A356991 1) for k >= 3, a(4*F(k-1) + j) = L(k) for 0 <= j <= F(k-4), where F(-1) = 1 and
%F A356991 2) for k >= 3, a(F(k+2) + j) = L(k) + j for 0 <= j <= L(k-1).
%p A356991 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 A356991 seq( b(n) + b(n - b(n) ), n = 2..100);
%Y A356991 Cf. A000032, A000045, A022087, A356988, A356992 - A356999.
%K A356991 nonn,easy
%O A356991 2,1
%A A356991 _Peter Bala_, Sep 08 2022