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Topic: sequence (Read 374 times) |
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tony123
Junior Member
Posts: 61
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K(n+2)=3k(n+1) - K(n) , k1=20
sequence 20,30,70,180,470,1230,...
find all natural numbers
such that
1+5K(n) k(n+1) is a perfect squer
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srn437
Newbie
the dark lord rises again....
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Re: sequence
« Reply #1 on: Aug 27th, 2007, 10:48am » |
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Is infinity a natural number?
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pex
Uberpuzzler
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Re: sequence
« Reply #2 on: Aug 27th, 2007, 11:04am » |
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on Aug 27th, 2007, 10:48am, srn347 wrote:| Is infinity a natural number? |
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No.
A few questions for Tony: are K(n) and k(n) the same? And is K(n) k(n+1) supposed to be multiplication? Is there any significance to you posting only the single initial value k1 (I suppose this is K(1)), while the recursion is of second degree?
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ThudnBlunder
wu::riddles Moderator
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Re: sequence
« Reply #3 on: Aug 27th, 2007, 12:27pm » |
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A start:
k(n) = 10* (2n-1), the 2n-1th Lucas number
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| « Last Edit: Aug 27th, 2007, 2:47pm by ThudnBlunder » |
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Eigenray
wu::riddles Moderator
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Re: sequence
« Reply #4 on: Aug 27th, 2007, 1:53pm » |
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Numerically we might note
(1 + 5knkn+1)1/2 ~ kn+kn+1.
In fact,
1 + 5knkn+1 = (kn+kn+1)2 + 501.
Proof: kn satisfies a second-order recurrence with characteristic polynomial x2 - 3x + 1, whose roots are r, r' = (3   5)/2, so kn is a linear combination of rn and r'n. Therefore both sides are linear combinations of r2n, r'2n, and (rr')n=1, so they both satisfy the same third-order recurrence (with characteristic polynomial (x-1)(x-r2)(x-r'2) = x3 - 8x2 + 8x - 1). So it suffices to check both sides agree for n=1,2,3.
Now, x2 = y2 + 501 is impossible for x > (501+1)/2 = 251, so we only need to check n for which 1+5knkn+1  2512, i.e., n 3, and we find n=3 as the only solution: 1 + 5k3k4 = (k3+k4)2 + 501 = 2502 + 501 = 2512.
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