
EXAMPLE

The sequence starts (8,7,6,5,4,3,2,1,0,8,8,8,8,8,8,8,8,8,7,6,5,4,3,2,1,0,8,8,...)
Instead of underlining terms, we will put parentheses around the terms we want to emphasize:
a(1) = 8 produces parentheses around a(1 + 9 = 10):
8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,7,6,5,4,3,2,1,0,8,8,...
a(2) = 7 produces parentheses around a(2 + 8 = 10), which is now already done. Then,
a(3) = 6 produces parentheses around a(3 + 7 = 10), which is already done. Then,
a(4) = 5 produces parentheses around a(4 + 6 = 10)  already done. Then,
a(5) = 4 produces parentheses around a(5 + 5 = 10)  already done. Then,
a(6) = 3 produces parentheses around a(6 + 4 = 10)  already done. Then,
a(7) = 2 produces parentheses around a(7 + 3 = 10)  already done. Then,
a(8) = 1 produces parentheses around a(8 + 2 = 10)  already done. Then,
a(9) = 0 produces parentheses around a(9 + 1 = 10)  already done. Then,
a(10) = 8 produces parentheses around a(10 + 9 = 19):
8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),6,5,4,3,2,1,0,8,8,...
a(11) = 8 produces parentheses around a(11 + 9 = 20):
8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),5,4,3,2,1,0,8,8,...
a(12) = 8 produces parentheses around a(12 + 9 = 21):
8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),4,3,2,1,0,8,8,...
a(13) = 8 produces parentheses around a(13 + 9 = 22):
8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),3,2,1,0,8,8,...
a(14) = 8 produces parentheses around a(14 + 9 = 23):
8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),(3),2,1,0,8,8,...
a(15) = 8 produces parentheses around a(15 + 9 = 24):
8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),(3),(2),1,0,8,8,...
a(16) = 8 produces parentheses around a(16 + 9 = 25):
8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),(3),(2),(1),0,8,8,...
a(17) = 8 produces parentheses around a(17 + 9 = 26):
8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),(3),(2),(1),(0),8,8,...
a(18) = 7 produces parentheses around a(18 + 8 = 26)  already done. Then,
a(19) = 6 produces parentheses around a(19 + 7 = 26)  already done. Etc.
We see in this small example that the parenthesized terms reproduce the initial sequence:
(8),(7),(6),(5),(4),(3),(2),(1),(0),...
The same is true for the subsequence of nonparenthesized terms:
8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 8, 8, 8, 8, 8, 8, 8,...
