

A272464


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


3



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

1,3


COMMENTS

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.
From Max Barrentine, Jul 29 2016: (Start)
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 oddindexed 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)


LINKS

Max Barrentine, Table of n, a(n) for n = 1..8204


FORMULA

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.


EXAMPLE

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...


PROG

(PARI) findprec(v) = {lastn = #v; forstep(k=lastn1, 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


CROSSREFS

Cf. A002487.
Van Eck's sequence A181391 has a somewhat similar flavor.  N. J. A. Sloane, May 03 2016
Sequence in context: A133084 A118851 A112383 * A133404 A134627 A064881
Adjacent sequences: A272461 A272462 A272463 * A272465 A272466 A272467


KEYWORD

nonn,tabf


AUTHOR

Max Barrentine, Apr 30 2016


EXTENSIONS

Definition revised by N. J. A. Sloane, May 03 2016


STATUS

approved



