

A005206


Hofstadter Gsequence: a(n)=na(a(n1)).
(Formerly M0436)


48



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 bottomtotop, lefttoright, one obtains the natural numbers: 1, 2, 3, 4, 5, 6, ... The tree has a recursive structure, since the following 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) = [ (n+1)*Phi]  [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
Rahman's abstract: "We give a combinatorial interpretation of a classical metaFibonacci sequence defined by G(n) = n  G(G(n1)) with the initial condition G(1) = 1, which appears in Hofstadter's 'Godel, Escher, Bach: An Eternal Golden Braid'. The interpretation is in terms of an infinite labelled tree. We then show a few corollaries about the behaviour of the sequence G(n) directly from the interpretation."  Jonathan Vos Post, May 09 2011
a(n) = n  A060144(n+1).  Reinhard Zumkeller, Apr 07 2012


REFERENCES

D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 137.
C. Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267273.
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)
M. Celaya, F. Ruskey, Morphic Words and Nested Recurrence Relations, arXiv preprint arXiv:1307.0153 [math.CO], 2013.
F. Michel Dekking, Pure morphic sequences and their standard forms, arXiv:1509.00260 [math.CO], 2015.
D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 3543.
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, 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
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], May 9, 2011.
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.  from 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"


FORMULA

a(n) = floor((n+1)*tau)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)=na(floor(n/tau)).  Benoit Cloitre, Nov 27 2002
a(n) = A019446(n)  1.  Reinhard Zumkeller, Feb 02 2012
a(n) = sum(A000045(m)*A213676(m,k): m=A000201(n+1), k=1..A072649(m)).  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


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]


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


CROSSREFS

Apart from initial terms, same as A060143. Cf. A123070.
a(n):=sum(h(k), k=1..n), n>=1, with h(k):= A005614(k1) and a(0):=0.
a(n)=A(n+1)(n+1), n>=0, with Wythoff numbers A(n):= A000201(n).
Cf. A060144, A019446, A072649, A213676.
Sequence in context: A057363 A073869 A060143 * A057365 A014245 A096386
Adjacent sequences: A005203 A005204 A005205 * A005207 A005208 A005209


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



