A005375 a(0) = 0; a(n) = n - a(a(a(a(n-1)))) for n > 0. (Formerly M0458)

0, 1, 1, 2, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 20, 21, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 30, 31, 31, 32, 32, 33, 34, 34, 35, 36, 37, 37, 38, 39, 40, 41, 41, 42, 42, 43, 44, 44, 45, 46, 47, 47, 48, 49, 50, 51, 51, 52, 53

Offset

0,4

Comments

Rule for n-th term: a(n) = An, where An denotes the Lamé antecedent to (or right shift of) n, which is found by replacing each Lm(i) (Lm(n) = Lm(n-1) + Lm(n-4): A003269) in the Zeckendorffian expansion (obtained by repeatedly subtracting the largest Lamé number you can until nothing remains) by Lm(i-1) (A1=1). For example: 58 = 50 + 7 + 1, so a(58)= 36 + 5 + 1 = 42. - Diego Torres (torresvillarroel(AT)hotmail.com), Nov 24 2002

a(A194081(n)) = n and a(m) <> n for m < A194081(n). - Reinhard Zumkeller, Aug 17 2011

From Pierre Letouzey, Feb 20 2025: (Start)

For all n >= 0, A005374(n) <= a(n) <= A005376(n) as proved in Letouzey-Li-Steiner link.

Last equality A005374(n) = a(n) for n = 18; last equality a(n) = A005376(n) for n = 25.

For all n >= 0, |a(n) - c*n| < 1.998 where c is the positive real root of x^4 + x - 1 = 0, c = 0.724491959000515611588372282... Proved in Letouzey link. (End)

References

D. Hofstadter, "Goedel, Escher, Bach", 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

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

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.

Index entries for Hofstadter-type sequences

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

Formula

a(n) = floor(c*n) + (-1) or 0 or 1 or 2, where c is the positive real root of x^4+x-1 = 0, c=0.724491959000515611588372282... (Conjectured with just 0 or 1 by Benoit Cloitre, Nov 05 2002; fixed and proved by Letouzey, see Letouzey link]. NB: see for instance a(120) = 88 for a difference of 2 and a(243) = 175 for a difference of -1). - Pierre Letouzey, Feb 20 2025

a(n + a(a(a(n)))) = n (proved in Letouzey-Li-Steiner link). - Pierre Letouzey, Feb 20 2025

Maple

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

Mathematica

a[0]:= 0; a[n_]:= a[n]= a[n] = n - a[a[a[a[n-1]]]]; Table[a[n], {n, 0, 73}] (* Alonso del Arte, Aug 17 2011 *)

Prog

(Haskell)
a005375 n = a005375_list !! n
a005375_list =  0 : 1 : zipWith (-)
   [2..] (map a005375 (map a005375 (map a005375 (tail a005375_list))))
-- Reinhard Zumkeller, Aug 17 2011
(SageMath)
@CachedFunction # a = A005375
def a(n): return 0 if (n==0) else n - a(a(a(a(n-1))))
[a(n) for n in range(101)] # G. C. Greubel, Nov 14 2022

Crossrefs

Sequence in context: A198454 A063882 A097873 * A138370 A125051 A064067

Adjacent sequences: A005372 A005373 A005374 * A005376 A005377 A005378

Keyword

nonn,nice,changed

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Jul 12 2000

Status

approved