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A272464
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Fractal sequence related to Stern's diatomic sequence (A002487).
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3
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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, 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, 21, 34, 13, 5, 2, 1, 4, 23, 19, 15, 41, 26, 37, 11, 40, 29, 47
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OFFSET
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1,3
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COMMENTS
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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).
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.
The "fractal" property is that if the first occurrence of each term that appears in this sequence is removed, the sequence remains unchanged.
For the above definition for n and k, does n/k converge to 3/2?
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)
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LINKS
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FORMULA
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For a(n)=a(k) for largest k<n, m<n:
If a(k)+a(k+1)≠a(m), then a(n+1)=a(k)+a(k+1), a(n+2)=a(k+1);
If a(k)+a(k+1)=a(m), then a(n+1)=a(k+1).
a(1)=a(2)=1.
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EXAMPLE
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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.
The terms may be displayed as a triangle, starting a new row when a 1 appears:
1;
1, 2;
1, 3, 2;
1, 4, 3, 5, 2;
1, 4, 7, 3, 8, 5, 2...
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PROG
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(PARI) findprec(v) = {lastn = #v; forstep(k=lastn-1, 1, -1, if (v[k] == v[lastn], return (k)); ); }
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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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