%I M0436 #239 Jun 11 2024 09:39:27

%S 0,1,1,2,3,3,4,4,5,6,6,7,8,8,9,9,10,11,11,12,12,13,14,14,15,16,16,17,

%T 17,18,19,19,20,21,21,22,22,23,24,24,25,25,26,27,27,28,29,29,30,30,31,

%U 32,32,33,33,34,35,35,36,37,37,38,38,39,40,40,41,42,42,43,43,44,45,45,46,46,47

%N Hofstadter G-sequence: a(0) = 0; a(n) = n - a(a(n-1)) for n > 0.

%C Rule for finding n-th term: a(n) = An, where An denotes the Fibonacci antecedent to (or right shift of) n, which is found by replacing each F(i) in the Zeckendorf expansion (obtained by repeatedly subtracting the largest Fibonacci number you can until nothing remains) by F(i-1) (A1=1). For example: 58 = 55 + 3, so a(58) = 34 + 2 = 36. - Diego Torres (torresvillarroel(AT)hotmail.com), Nov 24 2002

%C From Albert Neumueller (albert.neu(AT)gmail.com), Sep 28 2006: (Start)

%C A recursively built tree structure can be obtained from the sequence (see Hofstadter, p. 137):

%C 14 15 16 17 18 19 20 21

%C \ / / \ / \ / /

%C 9 10 11 12 13

%C \ / / \ /

%C 6 7 8

%C \ / /

%C \ / /

%C \ / /

%C 4 5

%C \ /

%C \ /

%C \ /

%C \ /

%C \ /

%C 3

%C /

%C 2

%C \ /

%C 1

%C To construct the tree: node n is connected with the node a(n) below

%C n

%C /

%C a(n)

%C For example, since a(7) = 4:

%C 7

%C /

%C 4

%C If the nodes of the tree are read from bottom to top, left to right, one obtains the positive integers: 1, 2, 3, 4, 5, 6, ... The tree has a recursive structure, since the construct

%C /

%C x

%C \ /

%C x

%C can be repeatedly added on top of its own ends, to construct the tree from its root: e.g.,

%C /

%C x

%C / \ /

%C x x

%C \ / /

%C x x

%C \ /

%C \ /

%C x

%C When moving from a node to a lower connected node, one is moving to the parent. Parent node of n: floor((n+1)/tau). Left child of n: floor(tau*n). Right child of n: floor(tau*(n+1))-1 where tau=(1+sqrt(5))/2. (See the Sillke link.)

%C (End)

%C The number n appears A001468(n) times; A001468(n) = floor((n+1)*Phi) - floor(n*Phi) with Phi = (1 + sqrt 5)/2. - _Philippe Deléham_, Sep 22 2005

%C Number of positive Wythoff A-numbers A000201 not exceeding n. - _N. J. A. Sloane_, Oct 09 2006

%C Number of positive Wythoff B-numbers < A000201(n+1). - _N. J. A. Sloane_, Oct 09 2006

%C From _Bernard Schott_, Apr 23 2022: (Start)

%C Properties coming from the 1st problem proposed during the 45th Czech and Slovak Mathematical Olympiad in 1996 (see IMO Compendium link):

%C -> a(n) >= a(n-1) for any positive integer n,

%C -> a(n) - a(n-1) belongs to {0,1},

%C -> No integer n exists such that a(n-1) = a(n) = a(n+1). (End)

%C For n >= 1, find n in the Wythoff array (A035513). a(n) is the number that precedes n in its row, using the preceding column of the extended Wythoff array (A287870) if n is at the start of the (unextended) row. - _Peter Munn_, Sep 17 2022

%C See my 2023 publication on Hofstadter's G-sequence for a proof of the equality of (a(n)) with the sequence A073869. - _Michel Dekking_, Apr 28 2024

%D D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 137.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H N. J. A. Sloane and T. D. Noe, <a href="/A005206/b005206.txt">Table of n, a(n) for n = 0..20000</a> (the first 1000 terms were found by T. D. Noe)

%H L. Carlitz, <a href="https://fq.math.ca/Scanned/6-4/carlitz.pdf">Fibonacci Representations</a>, Fibonacci Quarterly, volume 6, number 4, October 1968, pages 193-220. a(n) = e(n) at equation 1.10 or 2.11 and see equation 3.8 recurrence.

%H M. Celaya and F. Ruskey, <a href="http://arxiv.org/abs/1307.0153">Morphic Words and Nested Recurrence Relations</a>, arXiv preprint arXiv:1307.0153 [math.CO], 2013.

%H M. Celaya and F. Ruskey, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.121.06.549">Another Property of Only the Golden Ratio</a>, American Mathematical Monthly, Problem 11651, solutions volume 121, number 6, June-July 2014, pages 549-556.

%H F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

%H F. M. Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Dekking2/dek9.html">On Hofstadter's G-sequence</a>, Journal of Integer Sequences 26 (2023), Article 23.9.2, 1-11.

%H Larry Ericksen and Peter G. Anderson, <a href="http://www.cs.rit.edu/~pga/k-zeck.pdf">Patterns in differences between rows in k-Zeckendorf arrays</a>, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.

%H D. Gault and M. Clint, <a href="http://dx.doi.org/10.1080/00207168808803682">"Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function</a>, Internat. J. Computer Math., 26 (1988), 35-43. Also <a href="/A005206/a005206.pdf">annotated scanned copy</a>.

%H Martin Griffiths, <a href="https://www.fq.math.ca/Papers1/56-1/GriffithsmgFibWordSeq121517.pdf">A formula for an infinite family of Fibonacci-word sequences</a>, Fib. Q., 56 (2018), 75-80.

%H H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/15-4/gould.pdf">Sequences associated with t-ary coding of Fibonacci's rabbits</a>, Fib. Quart., 15 (1977), 311-318.

%H Vincent Granville and Jean-Paul Rasson, <a href="http://dx.doi.org/10.1016/0022-314X(88)90020-0">A strange recursive relation</a>, J. Number Theory 30 (1988), no. 2, 238--241. MR0961919(89j:11014).

%H J. Grytczuk, <a href="http://dx.doi.org/10.1016/j.disc.2003.10.022">Another variation on Conway's recursive sequence</a>, Discr. Math. 282 (2004), 149-161.

%H Nick Hobson, <a href="/A005206/a005206.py.txt">Python program for this sequence</a>

%H D. R. Hofstadter, <a href="/A006336/a006336_1.pdf">Eta-Lore</a> [Cached copy, with permission]

%H D. R. Hofstadter, <a href="/A006336/a006336_2.pdf">Pi-Mu Sequences</a> [Cached copy, with permission]

%H D. R. Hofstadter and N. J. A. Sloane, <a href="/A006336/a006336.pdf">Correspondence, 1977 and 1991</a>

%H The IMO Compendium, <a href="https://imomath.com/othercomp/Czs/CzsMO96.pdf">Problem 1</a>, 45th Czech and Slovak Mathematical Olympiad 1996.

%H Clark Kimberling and K. B. Stolarsky, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.3.267">Slow Beatty sequences, devious convergence, and partitional divergence</a>, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.

%H P. Letouzey, <a href="http://hal.inria.fr/hal-01195587">Hofstadter's problem for curious readers</a>, Technical Report, 2015.

%H Mustazee Rahman, <a href="http://arxiv.org/abs/1105.1718">A Combinatorial interpretation of Hofstadter's G-sequence</a>, arXiv:1105.1718 [math.CO], 2011-2013.

%H B. Schoenmakers, <a href="http://www.win.tue.nl/~berry/papers/lowskew.pdf">A tight lower bound for top-down skew heaps</a>, Information Processing Letters, 61(5): 279-284, 14 March 1997.

%H Torsten Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/floor-recurrence">Floor Recurrences</a>

%H Th. Stoll, <a href="http://www.fq.math.ca/Papers1/46_47-1/Stoll_11-08.pdf">On Hofstadter's married functions</a>, Fib. Q., 46/47 (2008/2009), 62-67. - _N. J. A. Sloane_, May 30 2009

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HofstadterG-Sequence.html">Hofstadter G-Sequence</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hofstadter_sequence">Hofstadter sequence</a>

%H <a href="/index/Ho#Hofstadter">Index entries for Hofstadter-type sequences</a>

%H <a href="/index/Go#GEB">Index entries for sequences from "Goedel, Escher, Bach"</a>

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.

%F a(n) = floor((n+1)*tau) - n - 1 = A000201(n+1)-n-1, where tau = (1+sqrt(5))/2; or a(n) = floor(sigma*(n+1)) where sigma = (sqrt(5)-1)/2.

%F a(0)=0, a(1)=1, a(n) = n - a(floor(n/tau)). - _Benoit Cloitre_, Nov 27 2002

%F a(n) = A019446(n) - 1. - _Reinhard Zumkeller_, Feb 02 2012

%F a(n) = n - A060144(n+1). - _Reinhard Zumkeller_, Apr 07 2012

%F a(n) = Sum_{k=1..A072649(m)} A000045(m)*A213676(m,k): m=A000201(n+1). - _Reinhard Zumkeller_, Mar 10 2013

%F a(n + a(n)) = n. - _Pierre Letouzey_, Sep 09 2015

%F a(n) + a(a(n+1) - 1) = n. - _Pierre Letouzey_, Sep 09 2015

%F a(0) = 0, a(n+1) = a(n) + d(n) and d(0) = 1, d(n+1)=1-d(n)*d(a(n)). - _Pierre Letouzey_, Sep 09 2015

%F a(n) = A293688(n)/(n+1) for n >= 0 (conjectured). - _Enrique Navarrete_, Oct 15 2017

%F A generalization of Diego Torres's 2002 comment as a formula: if n = Sum_{i in S} A000045(i+1), where S is a set of positive integers, then a(n) = Sum_{i in S} A000045(i). - _Peter Munn_, Sep 28 2022

%F Conjectures from _Chunqing Liu_, Aug 01 2023: (Start)

%F a(A000201(n)-1) = n-1.

%F a(A001950(n)-1) = a(A001950(n)) = A000201(n). (End)

%p H:=proc(n) option remember; if n=0 then 0 elif n=1 then 1 else n-H(H(n-1)); fi; end proc: seq(H(n),n=0..76);

%t a[0] = 0; a[n_] := a[n] = n - a[a[n - 1]]; Array[a, 77, 0]

%t (* Second program: *)

%t Fold[Append[#1, #2 - #1[[#1[[#2]] + 1 ]] ] &, {0}, Range@ 76] (* _Michael De Vlieger_, Nov 13 2017 *)

%o (Haskell)

%o a005206 n = a005206_list !! n

%o a005206_list = 0 : zipWith (-) [1..] (map a005206 a005206_list)

%o -- _Reinhard Zumkeller_, Feb 02 2012, Aug 07 2011

%o (Haskell)

%o a005206 = sum . zipWith (*) a000045_list . a213676_row . a000201 . (+ 1)

%o -- _Reinhard Zumkeller_, Mar 10 2013

%o (PARI) first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, v[k]=k-v[v[k-1]]); concat(0,v) \\ _Charles R Greathouse IV_, Sep 02 2015

%o (Magma) [Floor((n+1)*(1+Sqrt(5))/2)-n-1: n in [0..80]]; // _Vincenzo Librandi_, Nov 19 2016

%o (Python)

%o from math import isqrt

%o def A005206(n): return (n+1+isqrt(5*(n+1)**2)>>1)-n-1 # _Chai Wah Wu_, Aug 09 2022

%Y Apart from initial terms, same as A060143. Cf. A123070.

%Y a(n):=Sum{k=1..n} h(k), n >= 1, with h(k):= A005614(k-1) and a(0):=0.

%Y Cf. A060144, A019446, A072649, A213676, A000201.

%Y Cf. A035513, A287870.

%K nonn,easy,nice

%O 0,4

%A _N. J. A. Sloane_

%E a(0) = 0 added in the Name by _Bernard Schott_, Apr 23 2022