Can you Identify a Pattern in this Sequence?

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The discussion centers around identifying a pattern in a specific numerical sequence, which includes terms such as 1, 3, 5, 9, and extends to 5633. The user proposes a method of deriving terms based on differences of powers of two, specifically noting a pattern of differences like 2¹, 2², and 2³. However, the challenge remains in formulating a definitive recurrence relation that encapsulates the entire sequence. The user references the Frame-Stewart algorithm but seeks a more comprehensive pattern for analysis.

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Hello;

Can you identify a pattern in this sequence of numbers (a recurrence relation)?

1, 3, 5, 9, 13, 17, 25, 33, 41, 49, 65, 81, 97, 113, 129, 161, 193, 225, 257, 289, 321, 385, 449, 513, 577, 641, 705, 769, 897, 1025, 1153, 1281, 1409, 1537, 1665, 1793, 2049, 2305, 2561, 2817, 3073, 3329, 3585, 3841, 4097, 4609, 5121, 5633...

I can sort of see a pattern but not a recurrence relation... if we look at how we can progressively get from term to term, it looks like this:

2¹ + 2¹ + 2² + 2² + 2² + 2³ + 2³ + 2³ + 2³...

So whenever you end up with a difference of 2n, you add that difference to any term n+1 times.

But generating a recurrence relation is more difficult...

Can you see a pattern?
 
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http://www.research.att.com/~njas/sequences/A007664
 
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Hi;

Yes, I have looked at this page before and done a google search for my sequence too... But I already know about the Frame-Stewart algorithm and that page lists findings already described in my post. But I was just wondering if anyone here was able to find a better pattern that connects all the terms, such that a recurrence relation can be formed and solved.