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A227145 Numbers satisfying an infinite nested recurrence relation.

%I A227145 #29 Oct 12 2022 13:53:21
%S A227145 0,1,1,2,2,2,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,9,9,9,9,9,10,10,11,11,12,
%T A227145 12,13,13,13,14,14,15,15,15,16,16,17,17,17,17,18,18,19,19,20,20,20,21,
%U A227145 21,22,22,22,22,22,22,23,23,24,24,25,25,26,26,27,27,27
%N A227145 Numbers satisfying an infinite nested recurrence relation.
%C A227145 Conjecture: a(F_n) = F_{n-2} for n>1, where F_n is the n-th Fibonacci number.
%C A227145 Conjecture:  a(n) ~ n*(3-sqrt(5))/2. -_Jeffrey Shallit_, Oct 12 2022
%H A227145 Alois P. Heinz, <a href="/A227145/b227145.txt">Table of n, a(n) for n = 1..10000</a>
%H A227145 Marcel Celaya and Frank Ruskey, <a href="http://arxiv.org/abs/1307.0153">Morphic words and nested recurrence relations</a>, arxiv 1307.0153 (Jun 29 2013), [math.CO] (see page 11).
%F A227145 a(n) = n - 1 - a(n-1) - a(a(n-2)) - a(a(a(n-3))) - a(a(a(a(n-4)))) - ...  with a(n) = 0 if n <= 1.
%p A227145 a:= proc(n) option remember; local i, r, s;
%p A227145       if n<2 then 0 else r, s:= n, 1;
%p A227145          for i while s>0 do r, s:= r-s, (a@@i)(n-i) od: r
%p A227145       fi
%p A227145     end:
%p A227145 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jul 04 2013
%t A227145 a[n_] := a[n]= Which[n <= 1, 0,True, n - 1 -Sum[Nest[a, n - i, i], {i,1,n}]]; Table[a[i], {i, 0, 30}] (* _José María Grau Ribas_, Jul 10 2013 *)
%Y A227145 Cf. A060144.
%K A227145 nonn
%O A227145 1,4
%A A227145 _Frank Ruskey_, Jul 04 2013