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%I #11 Oct 13 2022 12:57:48
%S 0,1,1,0,1,1,1,1,2,2,2,1,1,2,3,3,3,3,3,2,2,2,3,4,5,5,5,5,5,5,5,4,3,3,
%T 3,3,4,5,6,7,8,8,8,8,8,8,8,8,8,8,8,7,6,5,5,5,5,5,5,6,7,8,9,10,11,12,
%U 13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,12,11,10,9,8,8,8,8,8,8,8,8,8,9,10,11
%N a(n) = b(n) - b(n - b(n - b(n))) for n >= 2, where b(n) = A356988(n).
%C The line graph of the sequence consists of a series of local plateaus and local troughs joined at each end by lines of slope 1 and slope -1. More precisely, for k >= 3 the graph of the sequence consists of
%C local plateaus: on the integer interval [2*F(k), 2*F(k) + 2*F(k-3)] the sequence has the constant value F(k-2)
%C descent to a trough: on the integer interval [2*F(k) + 2*F(k-3), F(k+2)] the line graph of the sequence has slope -1
%C local troughs: on the integer interval [F(k+2), F(k+2) + F(k-3)] the sequence has the constant value F(k-3)
%C ascent to a plateau: on the integer interval [F(k+2) + F(k-3), 2*F(k+1)] the line graph of the sequence has slope 1.
%F a(n+1) - a(n) = 1, 0 or -1.
%F Let F(n) = A000045(n) with F(-1) = 1 and let L(n) = A000032(n).
%F For k >= 5, a(F(k) + j) = F(k-5) for 0 <= j <= F(k-5) (troughs).
%F For k >= 4, a(2*F(k) + j) = F(k-2) for 0 <= j <= 2*F(k-3) (plateaus).
%e The sequence is arranged to show the local plateaus (P) and the local troughs (T):
%e 0,
%e 1,
%e 1,
%e T 0,
%e P 1, 1, 1
%e 1,
%e P 2, 2, 2,
%e T 1,1,
%e 2,
%e P 3, 3, 3, 3, 3,
%e T 2, 2, 2,
%e 3,
%e 4,
%e P 5, 5, 5, 5, 5, 5, 5,
%e 4,
%e T 3, 3, 3, 3,
%e 4,
%e 5,
%e 6,
%e 7,
%e P 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
%e 7,
%e 6,
%e T 5, 5, 5, 5, 5, 5,
%e 6,
%e 7,
%e 8,
%e 9,
%e 10,
%e 11,
%e 12,
%e P 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
%e 12,
%e 11,
%e 10,
%e 9,
%e T 8, 8, 8, 8, 8, 8, 8, 8, 8,
%e 9,
%e 10,
%e 11,
%e ...
%p # b(n) = A356988
%p 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 seq( b(n) - b(n - b(n - b(n))), n = 2
%p ..100);
%Y Cf. A000045, A356988, A356991 - A356999.
%K nonn,easy
%O 2,9
%A _Peter Bala_, Sep 11 2022
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