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A272464 Fractal sequence related to Stern's diatomic sequence (A002487).

%I A272464 #54 Aug 12 2016 06:22:02
%S A272464 1,1,2,1,3,2,1,4,3,5,2,1,4,7,3,8,5,2,1,4,11,7,10,3,8,13,5,2,1,4,15,11,
%T A272464 18,7,17,10,3,8,21,13,5,2,1,4,19,15,26,11,29,18,25,7,24,17,27,10,3,8,
%U A272464 21,34,13,5,2,1,4,23,19,15,41,26,37,11,40,29,47
%N A272464 Fractal sequence related to Stern's diatomic sequence (A002487).
%C A272464 To extend the sequence after a(n): suppose the most recent previous occurrence of a(n) was at a(k) (take the largest such k), if a(k)+a(k+1) does not occur earlier in the sequence, then we extend the sequence with two new terms, setting a(n+1)=a(k)+a(k+1) and a(n+2)=a(k+1); otherwise we get one new term by setting a(n+1)=a(k+1).
%C A272464 As a result of this construction, the last term added (except for the initial term) is always a term that has appeared before, and so k always exists.
%C A272464 The "fractal" property is that if the first occurrence of each term that appears in this sequence is removed, the sequence remains unchanged.
%C A272464 From _Max Barrentine_, Jul 29 2016: (Start)
%C A272464 For the above definition for n and k, does n/k converge to 3/2?
%C A272464 A different way to view this sequence is as Stern's Diatomic sequence (A002487) with repeating odd-indexed terms removed. For instance, A002487(6)=2, A002487(7)=3, and A002487(8)=1 but since 3 has already occurred in the present sequence at a(5), a(6)=2, and a(7)=1. This removes from the present sequence all terms that occur in A002487 between A002487(p)=2 and A002487(q)=1 for the largest p<q, or equivalently A002487(3*2^k) and A002487(2^k+1). (End)
%H A272464 Max Barrentine, <a href="/A272464/b272464.txt">Table of n, a(n) for n = 1..8204</a>
%F A272464 For a(n)=a(k) for largest k<n, m<n:
%F A272464 If a(k)+a(k+1)≠a(m), then a(n+1)=a(k)+a(k+1), a(n+2)=a(k+1);
%F A272464 If a(k)+a(k+1)=a(m), then a(n+1)=a(k+1).
%F A272464 a(1)=a(2)=1.
%e A272464 a(2)=1; the previous occurrence of a(n)=1 is a(1). Therefore a(3)=a(1)+a(2)=2, because 2 did not occur earlier in the sequence, and a(4)=a(2)=1.
%e A272464 The terms may be displayed as a triangle, starting a new row when a 1 appears:
%e A272464 1;
%e A272464 1, 2;
%e A272464 1, 3, 2;
%e A272464 1, 4, 3, 5, 2;
%e A272464 1, 4, 7, 3, 8, 5, 2...
%o A272464 (PARI) findprec(v) = {lastn = #v; forstep(k=lastn-1, 1, -1, if (v[k] == v[lastn], return (k)););}
%o A272464 lista(nn) = {v = [1, 1]; for (n= 1, nn, k = findprec(v); if (! vecsearch(vecsort(v,,8), v[k]+v[k+1]), v = concat(v, v[k]+v[k+1]); v = concat(v, v[k+1]), v = concat(v, v[k+1]));); print(v);} \\ _Michel Marcus_, May 02 2016
%Y A272464 Cf. A002487.
%Y A272464 Van Eck's sequence A181391 has a somewhat similar flavor. - _N. J. A. Sloane_, May 03 2016
%K A272464 nonn,tabf
%O A272464 1,3
%A A272464 _Max Barrentine_, Apr 30 2016
%E A272464 Definition revised by _N. J. A. Sloane_, May 03 2016