
EXAMPLE

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