cp's OEIS Frontend

A233526 Start with a(1) = 1, a(2) = 3, then a(n)*2^k = a(n+1) + a(n+2), with 2^k the smallest power of 2 (k>0) such that all terms a(n) are positive integers.

%I A233526 #29 May 22 2025 10:21:36
%S A233526 1,3,1,5,3,7,5,9,1,17,15,19,11,27,17,37,31,43,19,67,9,125,19,231,73,
%T A233526 389,195,583,197,969,607,1331,1097,1565,629,2501
%N A233526 Start with a(1) = 1, a(2) = 3, then a(n)*2^k = a(n+1) + a(n+2), with 2^k the smallest power of 2 (k>0) such that all terms a(n) are positive integers.
%C A233526 Define 2-free Fibonacci numbers as sequences where b(n) = (b(n-1) + b(n-2))/2^i such that 2^i is the greatest power of 2 that divides b(n-1) + b(n-2). Read backwards from the n-th term, this sequence produces a subsequence of 2-free Fibonacci numbers where we must divide by a power of 2 every time we add.
%C A233526 For other examples of n-free Fibonacci numbers, see A232666, A214684, A224382.
%H A233526 Brandon Avila, <a href="/A233526/b233526.txt">Table of n, a(n) for n = 1..1000</a>
%H A233526 Brandon Avila and Tanya Khovanova, <a href="http://arxiv.org/abs/1403.4614">Free Fibonacci Sequences</a>, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">J. Int. Seq. 17 (2014) # 14.8.5</a>.
%o A233526 (Python)
%o A233526 def minDivisionRich(n, a=1, b=3):
%o A233526     yield a
%o A233526     yield b
%o A233526     for i in range(2, n):
%o A233526         a *= 2
%o A233526         while a <= b:
%o A233526             a *= 2
%o A233526         a, b = b, a - b
%o A233526         yield b
%Y A233526 Cf. A233525.
%K A233526 nonn
%O A233526 1,2
%A A233526 _Brandon Avila_ and _Tanya Khovanova_, Dec 11 2013