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 December 21st, 2012, 02:54 PM #1 Senior Member   Joined: Dec 2012 Posts: 148 Thanks: 0 Fibonacci $\LARGE{ \sum_{i=1}^{n} F_{i} \cdot F_{i+1} \cdot F_{i+2} = ? } \\ \\ also \; \LARGE{ \sum_{i=1}^{k} F_{3i} = ? }$
 December 21st, 2012, 04:04 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Fibonacci They're both linear recurrences ("C-finite"); the first is a(n) = 4a(n-1) + 3a(n-2) - 9a(n-3) + 2a(n-4) + a(n-5) and the second is a(n) = 5a(n-1) - 3a(n-2) - a(n-3).
December 21st, 2012, 04:48 PM   #3
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Re: Fibonacci

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 Originally Posted by CRGreathouse They're both linear recurrences ("C-finite"); the first is a(n) = 4a(n-1) + 3a(n-2) - 9a(n-3) + 2a(n-4) + a(n-5) and the second is a(n) = 5a(n-1) - 3a(n-2) - a(n-3).
Ok. For 2nd: by a(n) you mean a(3k); a=3k

December 23rd, 2012, 02:22 PM   #4
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Re: Fibonacci

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 Originally Posted by tahir.iman Ok. For 2nd: by a(n) you mean a(3k); a=3k
I mean a(n) to be the sum of F(3k) from k = 1 to n. I don't mean a(3k).

 December 26th, 2012, 08:48 PM #5 Senior Member   Joined: Dec 2012 Posts: 148 Thanks: 0 Re: Fibonacci - I solved this... $\sum_{i=0}^{k} F_{ \tiny{3i+1} } = \frac{1}{2} \cdot F_{ \tiny{3k+3} } \\ \sum_{i=0}^{k} F_{ \tiny{3i+2} } = \frac{1}{2} \cdot ( F_{ \tiny{3k+4} } -1 ) \\ \sum_{i=1}^{k} F_{ \tiny{3i} } = \frac{1}{2} \cdot ( F_{ \tiny{3k+2} } -1 )$
 December 26th, 2012, 09:30 PM #6 Senior Member   Joined: Dec 2012 Posts: 148 Thanks: 0 Re: Fibonacci And also $\sum_{i=1}^{n+1} { F_{ \tiny{i} } }^{ \tiny{3} } = \frac{1}{2} \cdot ( F_{ \tiny{3n+2} } +1 ) - {F_{n} }^{ \tiny{3} }$
 December 27th, 2012, 05:20 AM #7 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Fibonacci You might enjoy the book A = B, in which such identities are called "routine".
December 27th, 2012, 07:27 AM   #8
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Re: Fibonacci

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 Originally Posted by CRGreathouse You might enjoy the book A = B, in which such identities are called "routine".
I dont waste my time with useless stuff. I advice this to you, also.

 December 27th, 2012, 07:32 AM #9 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Fibonacci What are you calling "useless", your identities or the book?
December 27th, 2012, 07:48 PM   #10
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Re: Fibonacci

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 Originally Posted by CRGreathouse What are you calling "useless", your identities or the book?
Book.

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