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%I #1 Jun 01 2010 03:00:00
%S 0,1,2,2,2,3,4,5,6,6,6,7,8,8,8,9,10,11,12,12,12,13,14,15,16,16,16,17,
%T 18,18,18,19,20,21,22,22,22,23,24,24,24,25,26,27,28,28,28,29,30,31,32,
%U 32,32,33,34,34,34,35,36,37,38,38,38,39,40,41,42,42,42,43,44,44,44,45,46
%N a(n)=n-a(a(n-2)) with a(0)=0,a(1)=1
%C A generalization of the Hofstadter G-sequence A005206 since it is part of the following family of sequences:
%C a(n)=n-a(a(n-k)) with the initial values a(0)=0,a(1)=a(2)=...=a(k-1)=1 and with k=1,2,3... (here k=2)
%C Every a(n) occurs either exactly one or exactly three times. Two blocks of three same elements are interrupted by either exactly one singular or exactly three consecutive natural numbers.
%C Since every natural number occurs in the sequence at least once 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 26 elements the following (ternary) tree:
%C ....1..............................
%C ....|..............................
%C ....2..............................
%C ./..|...\..........................
%C ....|......\.......................
%C ....|.........\....................
%C ....3...........4..................
%C ....|.............\................
%C ....5...............6..............
%C ....|.........../...|...\..........
%C ....7........8......9....10........
%C ....|....../.|.\....|.....\........
%C ....|...../..|..\...|......\.......
%C ....|..../....|..\..|.......\......
%C ...11...12....13.14.15......16.....
%C ....|../.|.\...|..|..|..../..|..\..
%C ...17.18.19.20.21.22.23.24..25..26.
%C Conjecture: Which features a certain structure (Comparable to A005206 or A135414). 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):
%C Diagram of D:
%C .....x......
%C .../.|.\....
%C ..D..C..x...
%C .........\..
%C ..........D.
%C Diagram of C:
%C ..x..
%C ..|..
%C ..C..
%Y Same recurrence relation as A135414.
%K easy,nonn
%O 0,3
%A Daniel Platt (d.platt(AT)web.de), Aug 04 2009