%I #5 Feb 22 2020 20:57:36
%S 0,0,0,0,4,5,6,7,4,4,4,4,8,9,10,11,12,13,14,15,12,12,12,12,16,17,18,
%T 19,16,16,16,16,20,21,22,23,24,25,26,27,24,24,24,24,28,29,30,31,32,33,
%U 34,35,32,32,32,32,36,37,38,39,36,36,36,36,40,41,42,43,44,45,46,47,44,44
%N a(n)=n-a(a(n-4)) with a(0)=a(1)=a(2)=a(3)=0.
%C 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):
%C 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 occurrences 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).
%C 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:
%C ..a..
%C ..|..
%C .a(n)
%C This will give for the first 55 elements the following (quintary) tree:
%C ..............................4...................
%C ...................../.../....|....\...\..........
%C .................../.../......|......\...\........
%C ......................8.......9.......10..11......
%C ..................../.........|........\....\.....
%C ..................12.........13.........14...15...
%C ................./...\\\\..../........../.../.....
%C ................/...__\_\\\_/........../.../......
%C .............../.../..__\_\_\\________/.../.......
%C ............./..../../.___\_\_\_\________/........
%C .........../...../.././....\.\.\..\...............
%C .........16.....17.18.19..20.21.22.23.............
%C ......../\\\\__/__/__/__...\..\..\..\.............
%C ......./..\\\_/__/__/_..\...\..\..\..\............
%C ....../....\\/__/__/_.\..\...\...\..\..\..........
%C ...../......X__/__/_.\.\..\...\...\..\..\.........
%C ..../....../../../..\.\.\..\...\....\..\..\.......
%C ...24....25.26.27..28.29.30.31.32....33.34.35.....
%C ../\\\\__/__/__/__...|.|..|.|..\\\\_/__/__/__.....
%C ./..\\\_/__/__/_..\..\.\..|.|..|\\\/__/__/__.\....
%C |....\\/__/__/_.\..\..\.\.|./..|.\X__/__/__.\.\...
%C |.....X__/__/_.\.\..\..\.\\/...|./\__|_|__.\.\.\..
%C |..../../../..\.\.\..\..\.\/...|.|...|.|..\.\.\.\.
%C 36.27.38.39..40.41.42.43..44..48.49.50.51.52.\54.\
%C ...........................45................53.55
%C ...........................46.....................
%C ...........................47.....................
%C (X means two crossing paths)
%C 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):
%C Diagram of D:
%C ......x.............
%C ..../..\\\\.........
%C .../....\\\.\.......
%C ..|......\\.\.\.....
%C ..|.......\.\.\.\...
%C ..|........\.\.\.\..
%C ..D..o.o.o..x.x.x.x.
%C ............|.|.|.|.
%C ............D.C.C.C.
%C (o will be filled by C)
%C Diagram of C:
%C \\\..x..
%C \\\\/...
%C .\\/\...
%C ../\\\..
%C ./.\\\\.
%C C...\\\\
%C (This means construct C crosses on its way from a(n) to n exactly four other paths, e.g. from 18 to 26)
%K nonn
%O 0,5
%A Daniel Platt (d.platt(AT)web.de), Aug 08 2009
%E Terrible typos here and in A163874 and A163873! Corrected the sequence definition. Two further changes will be requested soon. A thousand apologies for the inconvenience Daniel Platt (d.platt(AT)web.de), Sep 14 2009
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