0,4

Rule of construction for the sequence: a(n) = An, where An denotes the Lam{\'}e antecessor to (or right shift of) n, which is found by replacing each Lm(i) in the Zeckendorffian expansion (obtained by repeatedly subtracting the largest Lam{\'}e number (A000930) you can until nothing remains) by Lm(i-1) (A1=1). For example: 58 = 41 + 13 + 4, so a(58)= 28 + 9 + 3 = 40.

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

(Start) As is shown on page 137 of Goedel, Escher, Bach, a recursively built tree structure can be obtained from this sequence:

20.21..22..23.24.25.26.27.28

.\./.../.../...\./...\./../

..14.15..16....17....18..19

...\./.../..../.......\./

....10.11...12........13

.....\./.../........./

......7...8........9.

.......\./......./

........5......6

.........\.../

...........4

........../

.........3

......../

.......2

....\./

.....1

To construct the tree: node n is connected to the node a(n) below it:

...n

../

a(n)

For example:

...8

../

.5

since a(8) = 5. If the nodes of the tree are read from bottom-to-top, left-to-right, we obtain the natural numbers: 1, 2, 3, 4, 5, 6, ...

The tree has a recursive structure, since the following construct

....../

.....x

..../

...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

..\./...../

...x.....x

....\.../

......x (End)

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).

Nick Hobson, Python program for this sequence

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"

Conjecture: a(n) = floor(c*n) + 0 or 1, where c is the real root of x^3+x-1 = 0, c=0.682327803828019327369483739... - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 05 2002

Equals = A020942 - 2*A064105 + A064106 = 2*A020942 - A064105 - A001477. [Daniel Platt points out that there must be an error in this formula, since it fails for n=30: H(30)=20, A020942(30)=93, A064105(30)=131, A064106(30)=186, A001477(30)=30. Hence 20=93-2*131+186=2*93-131-30 <=> 20=17=25. Sep 11 2009]

Also: a(n) = a(n-1) + 1 if n-1 belongs to sequence A064105-A020942-A000012, a(n-1) otherwise.

Recurrence: a(n) = a(n-1) if n-1 belongs to sequence A020942, a(n-1) + 1 otherwise.

Recurrence for n>=3: a(n) = Lm(k-1) + a(n-Lm(k)), where Lm(n) denotes Lam{\'e} sequence A000930(n) (Lm(n) = Lm(n-1) + Lm(n-3)) and k is such that Lm(k)< n <= Lm(k+1). Special case: a(Lm(n)) = Lm(n-1) for n>=1.

A005374 := proc(n) option remember: if n<1 then 0 else n-A005374(A005374(A005374(n-1))) fi end: # from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Jan 06 2002

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

Sequence in context: A050292 A071521 A039733 this_sequence A071991 A096336 A081609

Adjacent sequences: A005371 A005372 A005373 this_sequence A005375 A005376 A005377

nonn,nice

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

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 12 2000

Additional comments and formulae from Diego Torres (torresvillarroel(AT)hotmail.com), Nov 23 2002