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Does it continue? 10: Alternate differences (Posted on 2017-10-07) Difficulty: 3 of 5
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

a1=1, an+1=[√(2an(an+1))] for n≥1 where [x] is the floor function.

Here are the first 17 terms and the alternate differences a2k+1 - a2k

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?

No Solution Yet Submitted by Jer    
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Some Thoughts Thoughts | Comment 1 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 2017-10-07 11:34:32
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