Binomial coefficient summatory and Fibonacci numbers question

olmoelisa

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