OFFSET
2,1
COMMENTS
The sequence is slow, that is, for n >= 2, a(n+1) - a(n) is either 0 or 1. The sequence is unbounded.
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.
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}.
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.
FORMULA
The sequence is completely determined by the initial values a(2) = 2, a(3) = 3 and the pair of formulas:
1) for k >= 3, a(4*F(k-1) + j) = L(k) for 0 <= j <= F(k-4), where F(-1) = 1 and
2) for k >= 3, a(F(k+2) + j) = L(k) + j for 0 <= j <= L(k-1).
MAPLE
b := proc(n) option remember; if n = 1 then 1 else n - b(b(n - b(b(b(n-1))))) end if; end proc:
seq( b(n) + b(n - b(n) ), n = 2..100);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Sep 08 2022
STATUS
approved