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[Core] Problem on sequence

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Problem on sequence

a sequence is defined as a(n+3)=a(n+2)+a(n+1)+a(n) and a(1)= 1, a(2) = 2, a(3) = 3.
Find a(2016)
   

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You should try and work out what the pattern in the series is. This is how I would go about doing it:
a(4) = a(1+3) = a(1+2) + a(1+1) +a(1) = a(3) + a(2) + a(1)
a(5) = a(4) + a(3) + a(2) = a(3) + a(2) + a(1) + a(3) + a(2) = 2a(3) + 2a(2) + a(1)
a(6) = a(5) + a(4) +a(3) = 2a(3) + 2a(2) + a(1) +a(4) + a(3) = 2a(3) + 2a(2) + a(1) + a(3) + a(2) +         
          a(1) + a(3) = 3a(3) + 3a(2) + 2a(1)
If you continue to work see what a(7) and a(8) are, you should then be able to work out a general pattern of a(m) = xa(3) + ya(2) + za(1), where m is an integer greater than or equal to 4, and x, y and z are integers.

Hope this helps.

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回覆 2# jackyjackychan 的帖子

x,y,z will form another tribonacci sequence. This is a vicious cycle.

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We can solve this problem using matrices. Harnessing eigenvalues and eigenvectors, we can find out the general term.

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Re ; marco
the eigenvalues are a real number and two complex numbers
how to find the corresponding eigenvectors
and in fact the real root are in ugly form.

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You have to admit that the general term is indeed "ugly".

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