This discrepancy appears in physics too. It's common to use 1,2,3 for spatial indices, but when you reach enlightenment and think in terms of spacetime you add a zero index and not a four.
That's only because you insist on explicitly mentioning the last element, which you can only do when the sequence is finite and non-empty (more generally, when it is indexed by a successor ordinal). So your choice of notation is not only inelegant, it cannot even express all possible sequences.
