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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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"

Programs

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

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 *)

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

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

Extensions

More terms from James Sellers, Jul 12 2000