0,4

Rule of construction for the sequence: a(n) = An, where An denotes the Fibonacci antecessor 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

Comments from Albert Neumueller (albert.neu(AT)gmail.com), Sep 28 2006.

"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 bottum-to-top, left-to-right, 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.)"

The number n appears A001468(n) times; A001468(n) = [ (n+1)*Phi] - [n*Phi] with Phi = (1+sqrt 5)/2 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 22 2005

Number of positive Wythoff A-numbers A000201 not exceeding n.

Number of positive Wythoff B-numbers < A000201(n+1).

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

D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43.

H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., Sequences associated with t-ary coding of Fibonacci's rabbits, Fib. Quart., 15 (1977), 311-318.

V. Granville and J. P. Rasson, A strange recursive relation, J. Number Theory, 30 (1988), 238-241.

J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.

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

Th. Stoll, On Hofstadter's married functions, Fib. Q., 46/47 (2008/2009), 62-67. - from N. J. A. Sloane, May 30 2009

T. D. Noe, Table of n, a(n) for n=0..1000

Nick Hobson, Python program for this sequence

B. Schoenmakers, A tight lower bound for top-down skew heaps, Information Processing Letters, 61(5): 279-284, 14 March 1997.

Torsten Sillke, Title?

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for Hofstadter-type sequences

Index entries for sequences from "Goedel, Escher, Bach"

a(n)=floor((n+1)*tau)-n-1 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 (benoit7848c(AT)orange.fr), Nov 27 2002

H:=proc(n) option remember; if n=1 then 1 else n-H(H(n-1)); fi; end proc;

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

a(n):=sum(h(k), k=1..n), n>=1, with h(k):= A005614((k-1) and a(0):=0.

a(n)=A(n+1)-(n+1), n>=0, with Wythoff numbers A(n):= A000201(n).

Sequence in context: A057363 A073869 A060143 this_sequence A057365 A014245 A096386

Adjacent sequences: A005203 A005204 A005205 this_sequence A005207 A005208 A005209

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).