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Binomial coefficient summatory and Fibonacci numbers question

  1. Nov 22, 2011 #1
    There is a summatory of binomial coefficients wich gives the Fibonacci

    numbers:


    (5 0) + (4 1) + (3 2) = 1 + 4 + 3 = 8 (Fib 7)

    (9 0) + (8 1) + (7 2) + (6 3) + (5 4) = 1 + 8 + 21 + 20 + 5 = 55 (Fib 10)

    If I alterne sum and subtraction I obtain 0, 1 or -1:

    1 - 4 + 3 = 0
    1 - 8 + 21 - 20 + 5 = -1

    But what happen if I sum the 1st half and subtract the 2nd half of the

    sequence?

    That is:

    1 + 4 -3 = 2 (or 1 - 4 -3 = - 6)
    1 + 8 + 21 - 20 - 5 = 5 (or 1 + 8 - 21 - 20 - 5 = -37)

    Any idea/paper/hint?

    Thank you very much
    Sandra
     
  2. jcsd
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