| A very near generalization of the Hofstadter G-sequence A005206 since it is part of the following family of sequences (which would give for k=1 the original G-sequence):
a(n)=n-a(a(n-k)) with a(0)=a(1)=...=a(k-1)=0 with k=1,2,3... (here k=4) - for general information about that family see A163873) Every a(n) occurs either exactly one or exactly five times (except from the initial values). A block of five occurences of the same number n is after the first one interrupted by the following three elements: n+1,n+2 and n+3 (e.g. see from a(16) to a(23): 12, 13, 14, 15, 12, 12, 12, 12).
Since every natural number occurs in the sequence at least once and 0<=a(n)<=n for all n the elements can be ordered in such a way that every n is connected to its a(n) in a tree structure so that:
..a..
..|..
.a(n)
This will give for the first 55 elements the following (quintary) tree:
..............................4...................
...................../.../....|....\...\..........
.................../.../......|......\...\........
......................8.......9.......10..11......
..................../.........|........\....\.....
..................12.........13.........14...15...
................./...\\\\..../........../.../.....
................/...__\_\\\_/........../.../......
.............../.../..__\_\_\\________/.../.......
............./..../../.___\_\_\_\________/........
.........../...../.././....\.\.\..\...............
.........16.....17.18.19..20.21.22.23.............
......../\\\\__/__/__/__...\..\..\..\.............
......./..\\\_/__/__/_..\...\..\..\..\............
....../....\\/__/__/_.\..\...\...\..\..\..........
...../......X__/__/_.\.\..\...\...\..\..\.........
..../....../../../..\.\.\..\...\....\..\..\.......
...24....25.26.27..28.29.30.31.32....33.34.35.....
../\\\\__/__/__/__...|.|..|.|..\\\\_/__/__/__.....
./..\\\_/__/__/_..\..\.\..|.|..|\\\/__/__/__.\....
|....\\/__/__/_.\..\..\.\.|./..|.\X__/__/__.\.\...
|.....X__/__/_.\.\..\..\.\\/...|./\__|_|__.\.\.\..
|..../../../..\.\.\..\..\.\/...|.|...|.|..\.\.\.\.
36.27.38.39..40.41.42.43..44..48.49.50.51.52.\54.\
...........................45................53.55
...........................46.....................
...........................47.....................
(X means two crossing pathes)
Conjecture: This features a certain structure (similar to the G-sequence A005206 or other sequences of this family: A163874 and A163873). If the (below) following two constructs (C and D) are added on top of their ends (either marked with C or D) one will (if starting with one instance of D) receive the above tree (x marks a node, o marks spaces for nodes that are not part of the construct but will be filled by the other construct):
Diagram of D:
......x.............
..../..\\\\.........
.../....\\\.\.......
..|......\\.\.\.....
..|.......\.\.\.\...
..|........\.\.\.\..
..D..o.o.o..x.x.x.x.
............|.|.|.|.
............D.C.C.C.
(o will be filled by C)
Diagram of C:
\\\..x..
\\\\/...
.\\/\...
../\\\..
./.\\\\.
C...\\\\
(This means construct C crosses on its way from a(n) to n exactly four other pathes, e.g. from 18 to 26)
| 