OFFSET
0,4
COMMENTS
From Pierre Letouzey, Mar 06 2025: (Start)
For all n >= 0, A005376(n) <= a(n) as proved in Letouzey-Li-Steiner link. Last equality a(n) = A005376(n) for n = 33. Moreover a(n) <= b(n) for all sequences b also defined by b(0)=0 and then b(n)=n-b(...b(n-1)...) with more than 6 nested recursive calls.
a(n) = c*n + O(n^d), where c is the real root of x^6+x-1 = 0, c=0.7780895986786012... and d=0.1287... Proved in Letouzey link. See also 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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
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. See p. 1.
Antoine Renard and Michel Rigo, Variants of Wythoff game with terminal positions or blocking maneuvers, Univ. Liège (2025). See p. 30.
FORMULA
a(n + a(a(a(a(a(n)))))) = n (proved in Letouzey-Li-Steiner link). - Pierre Letouzey, Mar 06 2025
MAPLE
H:=proc(n) option remember; if n=0 then 0 else n-H(H(H(H(H(H(n-1)))))); fi; end proc;
MATHEMATICA
a[0]= 0; a[n_]:= a[n]= n - a[a[a[a[a[a[n-1]]]]]]; Table[ a[n], {n, 75}] (* Robert G. Wilson v, Dec 16 2004 *)
PROG
(SageMath)
@CachedFunction # a = A100721
def a(n): return 0 if (n==0) else n - a(a(a(a(a(a(n-1))))))
[a(n) for n in range(1, 100)] # G. C. Greubel, Nov 16 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 12 2004
EXTENSIONS
a(0)=0 inserted by Pierre Letouzey, Mar 07 2025
STATUS
approved