A100721 a(n) = n - a(a(a(a(a(a(n-1)))))), a(0)=0.

0, 1, 1, 2, 3, 4, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 34, 34, 35, 35, 36, 37, 38, 39, 40, 41, 41, 42, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 50, 51, 51, 52, 53, 53, 54, 55, 56, 56, 57, 58, 59

Offset: 0

Author

N. J. A. Sloane, Dec 12 2004

Keywords

• nonn

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.

Crossrefs

Cf. A005206, A005374, A005375, A005376.

Programs

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 *)

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

Formula

a(n + a(a(a(a(a(n)))))) = n (proved in Letouzey-Li-Steiner link). - Pierre Letouzey, Mar 06 2025

Extensions

a(0)=0 inserted by Pierre Letouzey, Mar 07 2025