Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."
Consider the sequence
a_{1}=1, a_{n+1}=[√(2a_{n}(a_{n}+1))] for n≥1 where [x] is the floor function.
Here are the first 17 terms and the alternate differences a_{2k+1}  a_{2k}
1 2 3 4 6 9 13 19 27 38 54 77 109 154 218 309 437
1 2 4 8 16 32 64 128
Are they all powers of 2?
Initial thoughts  likely true, by analogy with similar series.
Consider a very similar sequence: a(n)=x, a(n+1)=[√(2(x)^2)], a(n+2) = [√(2([√(2(x)^2)])^2)]. But this is just 2x.
What's more interesting is explaining the odd values: 3,6,12+1,26+1,54,108+1,218,436+1,...

Posted by broll
on 20171007 11:34:32 