#powersof2 — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #powersof2, aggregated by home.social.
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Powers of 2 with all even digits
#HackerNews #PowersOf2 #EvenDigits #OEIS #A068994 #MathFun #NumberTheory
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Powers of 2 with all even digits
#HackerNews #PowersOf2 #EvenDigits #OEIS #A068994 #MathFun #NumberTheory
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Powers of 2 with all even digits
#HackerNews #PowersOf2 #EvenDigits #OEIS #A068994 #MathFun #NumberTheory
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Powers of 2 with all even digits
#HackerNews #PowersOf2 #EvenDigits #OEIS #A068994 #MathFun #NumberTheory
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Here's a very simple sequence (generalized from the Fibonacci sequence) to discourage students from generalizing a pattern too quickly. In fact, the sequence will look like it is the powers of 2 until it stops.
1, 1, 2, 4, 8, 16, ..., 2ᵏ, 2ᵏ⁺¹−1, 2ᵏ⁺²−3, 2ᵏ⁺³−8, ...
By selecting a detail in the sequence's (recursive) formula, I can control what the value of 𝑘 will be. So, technically, this is a family of sequences with the Fibonacci sequence being the one with 𝑘=2.
Reasons this family of sequences is cool:
1. I can control exactly what the value of the last power of 2 is and can make the pattern break after 2, 3, 10, 20, or 100 consecutive powers of 2 showing up.
2. The formula for this sequence is very easy to describe:
Start with a 1 and to find a new term, add up the last 𝑘 terms of the sequence (everything before the starting 1 that can be considered to be 0 if needed). Note that the 𝑘 terms being added up will match with the first 𝑘 powers of two (starting at 2⁰=1) showing up in the sequence before the pattern breaks.3. If you know the Fibonacci sequence (which is the special case of 𝑘=2), then this family of sequences is a natural generalization to look at. See:
https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Higher_orders4. If we adjust it to say "sum of all previous terms", we do in fact get the powers of two sequence.
Proof (by induction):
Base case: 1 + 1 = 2
Hypothesis: Assume that upto now, we've added up terms and gotten a power of two, say 2ᵏ.
Inductive step: For the next term, when we add all previous terms, we would add the terms that gave us 2ᵏ and then add the 2ᵏ term itself resulting in the sum of 2ᵏ⁺¹.#math; #pattern in a #sequence; #PowersOf2; #generalization of #Fibonacci.