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%I #54 Oct 30 2023 07:38:36
%S 6,0,164,0,60,2354,282,1336,100,1254,366,419,498,483,778,1204,292,373,
%T 845,838,1118,2120,815,2616,686,1195,745,1112,2132,1588,754,1227,1279,
%U 1661,716,2275,784,2341,1874,1463,1122,2800,1350,1613,2279,1557,1532
%N a(n) = length (or lifetime) of the meta-Fibonacci sequence {f(1) = ... = f(n) = 1; f(k)=f(k-f(k-1))+f(k-f(k-n))} if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.
%C Such a sequence has finite length when the k-th term becomes greater than k.
%C The term a(2) = 0 is only conjectural - see A005185. a(4) = 0 is a theorem of Balamohan et al. (2007). - _N. J. A. Sloane_, Nov 07 2007, Apr 18 2014.
%H Seiichi Manyama, <a href="/A134680/b134680.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H B. Balamohan, A. Kuznetsov and S. Tanny, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Tanny/tanny3.html">On the behavior of a variant of Hofstadter's Q-sequence</a>, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
%H D. R. Hofstadter, Curious patterns and non-patterns in a family of meta-Fibonacci recursions, Lecture in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, April 10 2014; <a href="https://vimeo.com/91708646">Part 1</a>, <a href="https://vimeo.com/91710600">Part 2</a>.
%H D. R. Hofstadter, <a href="/A134680/a134680.pdf">Graph of first 21500 terms</a>.
%H <a href="/index/Ho#Hofstadter">Index entries for Hofstadter-type sequences</a>
%e a(1) = 6: the f-sequence is defined by f(1) = 1, f(n) = 2f(n-f(n-1)), which gives 1,2,2,4,2,8 but f(7) = 2f(-1) is undefined, so the length is 6.
%t Table[Clear[a]; a[n_] := a[n] = If[n<=k, 1, a[n-a[n-1]]+a[n-a[n-k]]]; t={1}; n=2; While[n<10000 && a[n-1]<n, AppendTo[t,a[n]]; n++ ]; len=Length[t]; If[len==9999, 0, len], {k,100}]
%Y Cf. A005185, A046700, A063882, A132172, A134679 (sequences for n=2..6).
%Y See A240810 for another version.
%Y A diagonal of the triangle in A240813.
%K nonn
%O 1,1
%A _T. D. Noe_, Nov 06 2007