%I #17 Apr 30 2014 01:35:44
%S 1,1,2,3,4,4,4,5,6,6,7,8,9,9,9,10,11,12,12,12,13,14,14,15,16,17,17,17,
%T 18,19,19,20,21,22,22,22,23,24,25,25,25,26,27,27,28,29,30,30,30,31,32,
%U 33,33,33,34,35,35,36,37,38,38,38,39,40,40,41,42,43,43,43,44,45,46,46
%N a(1)=a(2)=1 and for n>=3, a(n)=n-a(a(n-2)).
%C A generalization of Hofstadter's G-sequence.
%C Contribution from Daniel Platt (d.platt(AT)web.de), Jul 27 2009: (Start)
%C Conjecture: A recursively built tree structure can be obtained from the sequence:
%C .29.30.31.32.33.34.35.36.37.38.39.40.41.42.43.44.45..
%C ..|..\./...|..|...\.|./...|..|...\.|./...|..\./...|..
%C .18..19...20.21....22....23.24....25....26..27...28..
%C ..\...|.../...|.....\..../...|.....|.....\...|.../...
%C ...\..|../....|......\../....|.....|......\..|../....
%C .....12......13.......14....15....16........17.......
%C ......|........\......|...../......|.........|.......
%C ......|..........\....|.../........|.........|.......
%C ......8...............9...........10........11.......
%C ......|.................\......./............|.......
%C ......|...................\.../..............|.......
%C ......5.....................6................7.......
%C .........\..................|............./..........
%C ..............\.............|........../.............
%C ....................\.......|....../.................
%C ............................4........................
%C .........................../.........................
%C ..........................3..........................
%C ........................./...........................
%C ........................2............................
%C ......................./.............................
%C ......................1..............................
%C When constructing the tree node n is connected to node a(n) below:
%C ..n..
%C ..|..
%C .a(n)
%C Same procedure as for A005206. Reading the nodes bottom-to-top, left-to-right provides the natural numbers. The tree has a recursive structure: The following construct will give - added on top of its own ends - the above tree:
%C .............. ... .
%C ............./.../..
%C ............/.../...
%C . ... .....X...X....
%C ..\...\.../.../.....
%C ...\...\./.../......
%C ....X...X...X.......
%C .....\..|../........
%C ......\.|./.........
%C ........X...........
%C (End)
%H D. Platt, <a href="/A135414/b135414.txt">Table of n, a(n) for n=1..1999</a> [From Daniel Platt (d.platt(AT)web.de), Jul 27 2009]
%F a(n)=2+floor(n*phi)+floor((n+1)*phi)-floor((n+3)*phi) where phi=(sqrt(5)-1)/2
%F n = a(n) + a(a(n-2)) unless n = 2 or n = -3. - _Michael Somos_, Jun 30 2011
%e x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + 6*x^10 + ...
%t a[ n_] := 2 - Boole[ n==0] + Quotient[ n, GoldenRatio] + Quotient[ n + 1, GoldenRatio] - Quotient[ n + 3, GoldenRatio] (* _Michael Somos_, Jun 30 2011 *)
%o (PARI) a(n)=2+floor(n*(sqrt(5)-1)/2)+floor((n+1)*(sqrt(5)-1)/2)-floor((n+3)*(sqrt(5)-1)/2)
%o (PARI) {a(n) = local(g = (1 + sqrt(5)) / 2); 2 - (n==0) + n\g + (n + 1)\g - (n + 3)\g} /* _Michael Somos_, Jun 30 2011 */
%o (Haskell)
%o a135414 n = a135414_list !! (n-1)
%o a135414_list = 1 : 1 : zipWith (-) [3..] (map a135414 a135414_list)
%o -- _Reinhard Zumkeller_, Nov 12 2011
%Y Cf. A005206.
%K nonn
%O 1,3
%A _Benoit Cloitre_, Feb 17 2008, Feb 19 2008