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 A163875 a(n)=n-a(a(n-4)) with a(0)=a(1)=a(2)=a(3)=0. 4
 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, 19, 16, 16, 16, 16, 20, 21, 22, 23, 24, 25, 26, 27, 24, 24, 24, 24, 28, 29, 30, 31, 32, 33, 34, 35, 32, 32, 32, 32, 36, 37, 38, 39, 36, 36, 36, 36, 40, 41, 42, 43, 44, 45, 46, 47, 44, 44 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 paths) 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 paths, e.g. from 18 to 26) LINKS CROSSREFS Sequence in context: A046345 A004445 A174630 * A244586 A114546 A067471 Adjacent sequences:  A163872 A163873 A163874 * A163876 A163877 A163878 KEYWORD nonn AUTHOR Daniel Platt (d.platt(AT)web.de), Aug 08 2009 EXTENSIONS 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 STATUS approved

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