This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A100721 #29 Mar 15 2025 04:36:02
%S A100721 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,
%T A100721 21,22,22,23,24,25,26,27,28,28,29,30,31,32,33,34,34,35,35,36,37,38,39,
%U A100721 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
%N A100721 a(n) = n - a(a(a(a(a(a(n-1)))))), a(0)=0.
%C A100721 From _Pierre Letouzey_, Mar 06 2025: (Start)
%C A100721 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.
%C A100721 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)
%D A100721 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.
%H A100721 G. C. Greubel, <a href="/A100721/b100721.txt">Table of n, a(n) for n = 0..10000</a>
%H A100721 Pierre Letouzey, <a href="https://arxiv.org/abs/2502.12615">Generalized Hofstadter functions G,H and beyond: numeration systems and discrepancy</a>, arXiv:2502.12615 [cs.DM], 2025.
%H A100721 Pierre Letouzey, Shuo Li, and Wolfgang Steiner, <a href="https://arxiv.org/abs/2410.00529">Pointwise order of generalized Hofstadter functions G, H and beyond</a>, arXiv:2410.00529 [cs.DM], 2024. See p. 1.
%F A100721 a(n + a(a(a(a(a(n)))))) = n (proved in Letouzey-Li-Steiner link). - _Pierre Letouzey_, Mar 06 2025
%p A100721 H:=proc(n) option remember; if n=0 then 0 else n-H(H(H(H(H(H(n-1)))))); fi; end proc;
%t A100721 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 *)
%o A100721 (SageMath)
%o A100721 @CachedFunction # a = A100721
%o A100721 def a(n): return 0 if (n==0) else n - a(a(a(a(a(a(n-1))))))
%o A100721 [a(n) for n in range(1,100)] # _G. C. Greubel_, Nov 16 2022
%Y A100721 Cf. A005206, A005374, A005375, A005376.
%K A100721 nonn
%O A100721 0,4
%A A100721 _N. J. A. Sloane_, Dec 12 2004
%E A100721 a(0)=0 inserted by _Pierre Letouzey_, Mar 07 2025