Originally posted by iamatiger
If you pre-seed 6 initial null elements with 0,0,0,0,0,1 then you can generate the number of ways of rolling each number as the sum of the previous 6 entries.
It looks to me that the positions of odd elements in this series are predictable, elements (0,1) are odd then elements (7,8), then elements (14,15).
These number are achievable with an odd numb ...[text shortened]... ions of rolls, therefore the ways of throwing them with odd rolls and even rolls cannot be equal
similarly, we can denote odd even pairs by (x,y) where x is the even entry and y is the odd entry, there are an even number of ways of throwing zero, and an odd number of ways of throwing one, and each later even entry is the sum of the six previous odd entries and vice versa, so:
It is clear that all totals which are multiples of 7 will have one more even way of throwing the number than odd ways, and all totals which are one more than a multiple of 7 will have one more odd way than even ways.