A005374 Hofstadter H-sequence: a(n) = n - a(a(a(n-1))).

0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17, 18, 18, 19, 20, 20, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 41, 41, 42, 42, 43, 44, 45, 45, 46, 46, 47, 48, 48, 49, 50

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Rule for constructing the sequence: a(n) = An, where An denotes the Lamé 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 Lamé 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.

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)

From Pierre Letouzey, Feb 20 2025: (Start)

For all n >= 0, A005206(n) <= a(n) <= A005375(n), as proved in Letouzey-Li-Steiner link. Last equality A005206(n) = a(n) occurs at n = 12; last equality a(n) = A005375(n) occurs at n = 18.

For all n >= 0, |a(n)-c*n| < 0.996, where c is the real root of x^3 + x - 1 = 0, c = 0.682327803828019327369483739... Proved in Letouzey link. (End)

The bound for |a(n)-c*n| is improved to 0.862 in Shallit (2025). - Jeffrey Shallit, Mar 09 2025

References

• 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

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Benoit Cloitre, Plot of a(n)-c*n where c=0.6823278...
• Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
• Nick Hobson, Python program for this sequence
• D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
• D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
• D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
• Pierre Letouzey, S. Li and W. Steiner, Pointwise order of generalized Hofstadter functions G,H and beyond, arXiv:2410.00529 [cs.DM], 2024.
• Pierre Letouzey, Generalized Hofstadter functions G,H and beyond: numeration systems and discrepancy arXiv:2502.12615 [cs.DM], 2025.
• Programming Puzzles & Code Golf Stack Exchange, Hofstadter H-sequence
• Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.
• Eric Weisstein's World of Mathematics, Hofstadter H-Sequence
• Wikipedia, Hofstadter sequence
• Index entries for Hofstadter-type sequences
• Index entries for sequences from "Goedel, Escher, Bach"

Crossrefs

Cf. A202340, A202341, A202342.

Programs

Haskell

a005374 n = a005374_list !! n
a005374_list = 0 : 1 : zipWith (-)
   [2..] (map (a005374 . a005374) $ tail a005374_list)
-- Reinhard Zumkeller, Dec 17 2011

Maple

A005374 := proc(n) option remember: if n<1 then 0 else n-A005374(A005374(A005374(n-1))) fi end: # 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;

Mathematica

a[n_]:= a[n]= n - a[a[a[n-1]]]; a[0] = 0; Table[a[n], {n, 0, 73}] (* Jean-François Alcover, Jul 28 2011 *)

PARI

first(m)=my(v=vector(m));v[1]=1;for(i=2,m,v[i]=i-v[v[v[i-1]]]);concat([0],v) \\ Anders Hellström, Dec 07 2015

SageMath

@CachedFunction # a = A005374
def a(n): return 0 if (n==0) else n - a(a(a(n-1)))
[a(n) for n in range(101)] # G. C. Greubel, Nov 14 2022

Formula

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, Nov 05 2002 [Proved by Letouzey, see Letouzey link. - Pierre Letouzey, Feb 20 2025], [Also proved in Shallit (2025). - Jeffrey Shallit, Mar 09 2025]

a(n) = A020942(n) - 2*A064105(n) + A064106(n) (e.g. for n = 30 we get 20 = 93 - 2*137 + 201), and a(n) = 2*A020942(n) - A064105(n) - A023443(n) (e.g. for n = 30 we get 20 = 2*93 - 137 - 29). [Corrected by N. J. A. Sloane, Apr 29 2024 at the suggestion of A.H.M. Smeets.]

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 Lamé 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.

For n > 0: a(A136495(n)) = n. - Reinhard Zumkeller, Dec 17 2011

Extensions

More terms from James Sellers, Jul 12 2000

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