Kata

Normally, we decompose a number into binary digits by assigning it with powers of 2, with a coefficient of 0 or 1 for each term:

25 = 1*16 + 1*8 + 0*4 + 0*2 + 1*1

The choice of 0 and 1 is... not very binary. We shall perform the true binary expansion by expanding with powers of 2, but with a coefficient of 1 or -1 instead:

25 = 1*16 + 1*8 + 1*4 - 1*2 - 1*1

Now this looks binary.

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Given any positive number n, expand it using the true binary expansion, and return the result as an array, from the most significant digit to the least significant digit.

true_binary(25) == [1,1,1,-1,-1]

It should be trivial (the proofs are left as an exercise to the reader) to see that:

• Every odd number has infinitely many true binary expansions
• Every even number has no true binary expansions

Hence, n will always be an odd number, and you should return the least true binary expansion for any n.

Also, note that n can be very, very large, so your code should be very efficient.