Question 4
Some natural numbers are initially written on a line. At each step, first identify every adjacent pair in the current left-to-right order, then insert their sum between each pair. After 2013 steps, how many occurrences of 2013 are present if initially:
(a) the numbers are $1,1000$;
(b) the numbers are $1,2,\ldots,1000$ in increasing order?
(a) the numbers are $1,1000$;
(b) the numbers are $1,2,\ldots,1000$ in increasing order?