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

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

Offset: 1

Author

Max Barrentine, Apr 30 2016

Keywords

• nonn
• tabf

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 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 pA002487(3*2^k) and A002487(2^k+1). (End)

Examples

			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...
		

Links

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

Crossrefs

Cf. A002487.

Van Eck's sequence A181391 has a somewhat similar flavor. - N. J. A. Sloane, May 03 2016

Programs

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

Formula

For a(n)=a(k) for largest k

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.

Extensions

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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