cp's OEIS Frontend

A356999 a(n) = 2*A356988(n) - n.

%I A356999 #13 Jan 04 2023 03:08:03
%S A356999 1,0,1,2,1,2,3,2,3,4,5,4,3,4,5,6,7,8,7,6,5,6,7,8,9,10,11,12,13,12,11,
%T A356999 10,9,8,9,10,11,12,13,14,15,16,17,18,19,20,21,20,19,18,17,16,15,14,13,
%U A356999 14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,33,32
%N A356999 a(n) = 2*A356988(n) - n.
%C A356999 For k >= 3, the line graph of the sequence consists of a series of local peaks of height Fibonacci(k) at abscissa n = Lucas(k) separated by local valleys of height Fibonacci(k-1) at abscissa n = Fibonacci(k+2).
%F A356999 a(n+1) - a(n) is either 1 or -1.
%F A356999 The sequence is completely determined by the pair of formulas
%F A356999 1) for k >= 2, a(F(k) + j) = F(k-3) + j for 0 <= j <= F(k-2) and
%F A356999 2) for k >= 1, a(L(k) + j) = F(k) - j for 0 <= j <= F(k-2),
%F A356999 where F(n) = A000045(n) denotes the n-th Fibonacci number (with F(-1) = 1) and L(n) = A000032(n) denotes the n-th Lucas number.
%e A356999 The sequence arranged as a series of alternating monotone increasing and decreasing sequences:
%e A356999   1;
%e A356999   0;
%e A356999   1, 2;
%e A356999   1;
%e A356999   2, 3;
%e A356999   2;
%e A356999   3, 4, 5;
%e A356999   4, 3;
%e A356999   4, 5, 6, 7, 8;
%e A356999   7, 6, 5;
%e A356999   6, 7, 8, 9, 10, 11, 12, 13;
%e A356999   12, 11, 10, 9, 8;
%e A356999   9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21;
%e A356999   20, 19, 18, 17, 16, 15, 14, 13;
%e A356999   14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, ..., 32, 33, 34;
%e A356999   33, 32, 31, ..., 23, 22, 21;
%p A356999 # b(n) = A356988(n)
%p A356999 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 A356999 seq(2*b(n) - n, n = 1..100);
%Y A356999 Cf. A000032, A000045, A356988, A356991 - A356998.
%K A356999 nonn,easy
%O A356999 1,4
%A A356999 _Peter Bala_, Sep 13 2022