A005376 - OEIS

%I M0464 #40 Apr 29 2024 13:46:45

%S 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,

%T 21,21,22,23,24,25,26,26,27,27,28,29,30,31,32,32,33,33,34,35,35,36,37,

%U 38,39,40,40,41,41,42,43,43,44,45,46,46,47,48,49,50,51,51,52,52,53,54,54

%N a(n) = n - a(a(a(a(a(n-1))))).

%C 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

%C 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

%D Douglas R. Hofstadter, "Goedel, Escher, Bach", p. 137.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A005376/b005376.txt">Table of n, a(n) for n = 0..10000</a>

%H Larry Ericksen and Peter G. Anderson, <a href="http://www.cs.rit.edu/~pga/k-zeck.pdf">Patterns in differences between rows in k-Zeckendorf arrays</a>, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.

%H Nick Hobson, <a href="/A005376/a005376.py.txt">Python program for this sequence</a>

%H <a href="/index/Ho#Hofstadter">Index entries for Hofstadter-type sequences</a>

%H <a href="/index/Go#GEB">Index entries for sequences from "Goedel, Escher, Bach"</a>

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

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

%t Table[a[n], {n, 0, 100}] (* _G. C. Greubel_, Nov 16 2022 *)

%o (SageMath)

%o @CachedFunction # a = A005376

%o def a(n): return 0 if (n==0) else n - a(a(a(a(a(n-1)))))

%o [a(n) for n in range(101)] # _G. C. Greubel_, Nov 16 2022

%Y Cf. A005206, A005374, A005375, A100721.

%K nonn

%O 0,4

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Jul 12 2000