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1, 2, 6, 42, ... |
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Topic: 1, 2, 6, 42, ... (Read 2021 times) |
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cool_joh
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an = an-1 (an-1 + 1)
How can we find the common formula for an?
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towr
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Re: 1, 2, 6, 42, ...
« Reply #1 on: Nov 23rd, 2007, 12:15am » |
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If you mean a closed formula, that may be tricky. I can't find a general method for solving these things (see also http://mathworld.wolfram.com/QuadraticMap.html )
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Wikipedia, Google, Mathworld, Integer sequence DB
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Barukh
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Re: 1, 2, 6, 42, ...
« Reply #2 on: Nov 23rd, 2007, 1:11am » |
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on Nov 23rd, 2007, 12:15am, towr wrote:| I can't find a general method for solving these things |
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It's available here.
I find the following very beautiful (taken from Sloane A000058):
"The greedy Egyptian representation of 1 is 1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + ... "
which is the sum of reciprocals of an + 1.
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cool_joh
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on Nov 23rd, 2007, 1:11am, Barukh wrote:
It's too complicated. I'm not good in English, can someone help me please?
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Barukh
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Re: 1, 2, 6, 42, ...
« Reply #4 on: Nov 23rd, 2007, 9:08am » |
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Bottom line is this: there exist a constant r, such that an equals to the nearest integer of r2^n (sequence starts at n=0). For this particular sequence, r equals 1.5979...
That's kind of cheating since to compute r one needs to know all an-s upfront (formula (19) in the article). The series converges very rapidly, though.
Looks like it's the best that could be done (or is known).
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