A005376 - OEIS

A005376

a(n) = n - a(a(a(a(a(n-1))))).
(Formerly M0464)

0, 1, 1, 2, 3, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 38, 39, 40, 40, 41, 41, 42, 43, 43, 44, 45, 46, 46, 47, 48, 49, 50, 51, 51, 52, 52, 53, 54, 54

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Conjecture: a(n) is approximately c*n, where c is the real root of x^5+x-1 = 0, c=0.754877666246692760049508896... - Benoit Cloitre, Nov 05 2002

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-5): A003520) in the Zeckendorffian expansion (obtained by repeatedly subtracting the largest Lamé number you can until nothing remains) with Lm(i-1) (A1=1). For example: 58 = 45 + 11 + 2, so a(58) = 34 + 8 + 1 = 43. - Diego Torres (torresvillarroel(AT)hotmail.com), Nov 24 2002

From Pierre Letouzey, Mar 6 2025: (Start)

For all n >= 0, A005375(n) <= a(n) <= A100721(n) as proved in Letouzey-Li-Steiner link. Last equality A005375(n) = a(n) for n = 25; last equality a(n) = A100721(n) for n = 33.

a(n) = c*n + O(ln(n)), with c conjectured by Benoit Cloitre above; see Letouzey link and Dilcher 1993. (End)

REFERENCES

Karl Dilcher, On a class of iterative recurrence relations, in G. E. Bergum, A. N. Philippou, and A. F. Horadam, editors, Applications of Fibonacci Numbers, vol. 5, p. 143-158, Springer, 1993.

Douglas R. 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

G. C. Greubel, 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, Generalized Hofstadter functions G,H and beyond: numeration systems and discrepancy, arXiv:2502.12615 [cs.DM], 2025.

Pierre Letouzey, Shuo Li and Wolfgang Steiner, Pointwise order of generalized Hofstadter functions G,H and beyond, arXiv:2410.00529 [cs.DM], 2024.

Index entries for Hofstadter-type sequences

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

FORMULA

a(n + a(a(a(a(n))))) = n (proved in Letouzey-Li-Steiner link). - Pierre Letouzey, Mar 6 2025

MAPLE

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

MATHEMATICA

a[n_]:= a[n]= If[n<1, 0, n -a[a[a[a[a[n-1]]]]]];

Table[a[n], {n, 0, 100}] (* G. C. Greubel, Nov 16 2022 *)