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A005376
a(n) = n - a(a(a(a(a(n-1))))).
(Formerly M0464)
5
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
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
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.
Pierre Letouzey, Shuo Li and Wolfgang Steiner, Pointwise order of generalized Hofstadter functions G,H and beyond, arXiv:2410.00529 [cs.DM], 2024.
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 *)
PROG
(SageMath)
@CachedFunction # a = A005376
def a(n): return 0 if (n==0) else n - a(a(a(a(a(n-1)))))
[a(n) for n in range(101)] # G. C. Greubel, Nov 16 2022
CROSSREFS
KEYWORD
nonn,changed
EXTENSIONS
More terms from James A. Sellers, Jul 12 2000
STATUS
approved