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A093879 First differences of A004001.

%I A093879 #28 May 20 2024 02:32:28
%S A093879 0,1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,1,1,1,
%T A093879 1,1,0,1,1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0,0,1,0,0,0,0,0,1,1,1,1,1,
%U A093879 1,0,1,1,1,1,0,1,1,1,0,1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0,0
%N A093879 First differences of A004001.
%C A093879 All the terms are 0 or 1: it is easy to show that if {b(n)} = A004001, b(n)>=b(n-1) and b(n)<n, therefore the first differences form an infinite binary word. - _Benoit Cloitre_, Jun 05 2004
%H A093879 R. J. Mathar, <a href="/A093879/b093879.txt">Table of n, a(n) for n = 1..9999</a>
%H A093879 J. Grytczuk, <a href="http://dx.doi.org/10.1016/j.disc.2003.10.022">Another variation on Conway's recursive sequence</a>, Discr. Math. 282 (2004), 149-161.
%H A093879 D. Newman, <a href="http://www.jstor.org/stable/2322766">Problem E3274</a>, Amer. Math. Monthly, 95 (1988), 555.
%F A093879 From _Antti Karttunen_, Jan 18 2016: (Start)
%F A093879 a(n) = A004001(n+1) - A004001(n).
%F A093879 Other identities. For all n >= 1:
%F A093879 a(A087686(n+1)-1) = 0.
%F A093879 a(A088359(n)-1) = 1.
%F A093879 a(n) = 1 if and only if A051135(n+1) = 1.
%F A093879 (End)
%t A093879 a[1] = a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; t = Table[a[n], {n, 110}]; Drop[t, 1] - Drop[t, -1] (* _Robert G. Wilson v_, May 28 2004 *)
%o A093879 (PARI) {m=106;v=vector(m,j,1);for(n=3,m,a=v[v[n-1]]+v[n-v[n-1]];v[n]=a);for(n=2,m,print1(v[n]-v[n-1],","))}
%o A093879 (Scheme) (define (A093879 n) (- (A004001 (+ 1 n)) (A004001 n))) ;; Code for A004001 given in that entry. - _Antti Karttunen_, Jan 18 2016
%o A093879 (Magma)
%o A093879 h:=[n le 2 select 1 else Self(Self(n-1)) + Self(n-Self(n-1)): n in [1..160]];
%o A093879 A093879:= func< n | h[n+1] - h[n] >;
%o A093879 [A093879(n): n in [1..120]]; // _G. C. Greubel_, May 19 2024
%o A093879 (SageMath)
%o A093879 @CachedFunction
%o A093879 def h(n): return 1 if (n<3) else h(h(n-1)) + h(n - h(n-1))
%o A093879 def A093879(n): return h(n+1) - h(n)
%o A093879 [A093879(n) for n in range(1,101)] # _G. C. Greubel_, May 19 2024
%Y A093879 Cf. A004001, A051135, A087686, A088359, A188163.
%K A093879 nonn,easy
%O A093879 1,1
%A A093879 _N. J. A. Sloane_, May 27 2004
%E A093879 More terms and PARI code from _Klaus Brockhaus_ and _Robert G. Wilson v_, May 27 2004