

A005206


Hofstadter Gsequence: a(0) = 0; a(n) = n  a(a(n1)) for n > 0.
(Formerly M0436)


72



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

0,4


COMMENTS

Rule for finding nth 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(i1) (A1=1). For example: 58 = 55 + 3, so a(58) = 34 + 2 = 36.  Diego Torres (torresvillarroel(AT)hotmail.com), Nov 24 2002
From Albert Neumueller (albert.neu(AT)gmail.com), Sep 28 2006: (Start)
A recursively built tree structure can be obtained from the sequence (see Hofstadter, p. 137):
14 15 16 17 18 19 20 21
\ / / \ / \ / /
9 10 11 12 13
\ / / \ /
6 7 8
\ / /
\ / /
\ / /
4 5
\ /
\ /
\ /
\ /
\ /
3
/
2
\ /
1
To construct the tree: node n is connected with the node a(n) below
n
/
a(n)
For example, since a(7) = 4:
7
/
4
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
/
x
\ /
x
can be repeatedly added on top of its own ends, to construct the tree from its root: e.g.,
/
x
/ \ /
x x
\ / /
x x
\ /
\ /
x
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.)
(End)
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
Number of positive Wythoff Anumbers A000201 not exceeding n.  N. J. A. Sloane, Oct 09 2006
Number of positive Wythoff Bnumbers < A000201(n+1).  N. J. A. Sloane, Oct 09 2006
From Bernard Schott, Apr 23 2022: (Start)
Properties coming from the 1st problem proposed during the 45th Czech and Slovak Mathematical Olympiad in 1996 (see IMO Compendium link):
> a(n) >= a(n1) for any positive integer n,
> a(n)  a(n1) belongs to {0,1},
> No integer n exists such that a(n1) = a(n) = a(n+1). (End)
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


REFERENCES

Martin Griffiths, A formula for an infinite family of Fibonacciword sequences, Fib. Q., 56 (2018), 7580.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 137.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

N. J. A. Sloane and T. D. Noe, Table of n, a(n) for n = 0..20000 (the first 1000 terms were found by T. D. Noe)
L. Carlitz, Fibonacci Representations, Fibonacci Quarterly, volume 6, number 4, October 1968, pages 193220. a(n) = e(n) at equation 1.10 or 2.11 and see equation 3.8 recurrence.
M. Celaya and F. Ruskey, Morphic Words and Nested Recurrence Relations, arXiv preprint arXiv:1307.0153 [math.CO], 2013.
M. Celaya and F. Ruskey, Another Property of Only the Golden Ratio, American Mathematical Monthly, Problem 11651, solutions volume 121, number 6, JuneJuly 2014, pages 549556.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 3543. Also annotated scanned copy.
H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., Sequences associated with tary coding of Fibonacci's rabbits, Fib. Quart., 15 (1977), 311318.
Vincent Granville and JeanPaul Rasson, A strange recursive relation, J. Number Theory 30 (1988), no. 2, 238241. MR0961919(89j:11014).
J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149161.
Nick Hobson, Python program for this sequence
D. R. Hofstadter, EtaLore [Cached copy, with permission]
D. R. Hofstadter, PiMu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
The IMO Compendium, Problem 1, 45th Czech and Slovak Mathematical Olympiad 1996.
Clark Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267273.
P. Letouzey, Hofstadter's problem for curious readers, Technical Report, 2015.
Mustazee Rahman, A Combinatorial interpretation of Hofstadter's Gsequence, arXiv:1105.1718 [math.CO], 20112013.
B. Schoenmakers, A tight lower bound for topdown skew heaps, Information Processing Letters, 61(5): 279284, 14 March 1997.
Torsten Sillke, Floor Recurrences
Th. Stoll, On Hofstadter's married functions, Fib. Q., 46/47 (2008/2009), 6267.  N. J. A. Sloane, May 30 2009
Eric Weisstein's World of Mathematics, Hofstadter GSequence
Wikipedia, Hofstadter sequence
Index entries for Hofstadtertype sequences
Index entries for sequences from "Goedel, Escher, Bach"
Index to sequences related to Olympiads.


FORMULA

a(n) = floor((n+1)*tau)  n  1 = A000201(n+1)n1, where tau = (1+sqrt(5))/2; or a(n) = floor(sigma*(n+1)) where sigma = (sqrt(5)1)/2.
a(0)=0, a(1)=1, a(n) = n  a(floor(n/tau)).  Benoit Cloitre, Nov 27 2002
a(n) = A019446(n)  1.  Reinhard Zumkeller, Feb 02 2012
a(n) = n  A060144(n+1).  Reinhard Zumkeller, Apr 07 2012
a(n) = Sum_{k=1..A072649(m)} A000045(m)*A213676(m,k): m=A000201(n+1).  Reinhard Zumkeller, Mar 10 2013
a(n + a(n)) = n.  Pierre Letouzey, Sep 09 2015
a(n) + a(a(n+1)  1) = n.  Pierre Letouzey, Sep 09 2015
a(0) = 0, a(n+1) = a(n) + d(n) and d(0) = 1, d(n+1)=1d(n)*d(a(n)).  Pierre Letouzey, Sep 09 2015
a(n) = A293688(n)/(n+1) for n >= 0 (conjectured).  Enrique Navarrete, Oct 15 2017
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


MAPLE

H:=proc(n) option remember; if n=0 then 0 elif n=1 then 1 else nH(H(n1)); fi; end proc: seq(H(n), n=0..76);


MATHEMATICA

a[0] = 0; a[n_] := a[n] = n  a[a[n  1]]; Array[a, 77, 0]
(* Second program: *)
Fold[Append[#1, #2  #1[[#1[[#2]] + 1 ]] ] &, {0}, Range@ 76] (* Michael De Vlieger, Nov 13 2017 *)


PROG

(Haskell)
a005206 n = a005206_list !! n
a005206_list = 0 : zipWith () [1..] (map a005206 a005206_list)
 Reinhard Zumkeller, Feb 02 2012, Aug 07 2011
(Haskell)
a005206 = sum . zipWith (*) a000045_list . a213676_row . a000201 . (+ 1)
 Reinhard Zumkeller, Mar 10 2013
(PARI) first(n)=my(v=vector(n)); v[1]=1; for(k=2, n, v[k]=kv[v[k1]]); concat(0, v) \\ Charles R Greathouse IV, Sep 02 2015
(Magma) [Floor((n+1)*(1+Sqrt(5))/2)n1: n in [0..80]]; // Vincenzo Librandi, Nov 19 2016
(Python)
from math import isqrt
def A005206(n): return (n+1+isqrt(5*(n+1)**2)>>1)n1 # Chai Wah Wu, Aug 09 2022


CROSSREFS

Apart from initial terms, same as A060143. Cf. A123070.
a(n):=Sum{k=1..n} h(k), n >= 1, with h(k):= A005614(k1) and a(0):=0.
Cf. A060144, A019446, A072649, A213676, A000201.
Cf. A035513, A287870.
Sequence in context: A057363 A073869 A060143 * A309077 A057365 A014245
Adjacent sequences: A005203 A005204 A005205 * A005207 A005208 A005209


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

a(0) = 0 added in the Name by Bernard Schott, Apr 23 2022


STATUS

approved



