<span style="font-family: courier new,monospace;">Rewriting Andrew's
numbers instead with, say, 4332 re-coded as 1210, that is, using the
place-value to represent whether the digits are 1, 2, 3, or 4, I get</span><br style="font-family: courier new,monospace;"><span style="font-family: courier new,monospace;">> 1 = 1 = 1</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">> 10 = 2 = 2</span><br style="font-family: courier new,monospace;"><span style="font-family: courier new,monospace;">> 11 = 21 = 3</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">> 20 = 22 = 4</span><br style="font-family: courier new,monospace;"><span style="font-family: courier new,monospace;">> 110 = 32 = 5</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">> 111 = 321 = 6</span><br style="font-family: courier new,monospace;"><span style="font-family: courier new,monospace;">> 120 = 322 = 7</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">> 210 = 332 = 8</span><br style="font-family: courier new,monospace;"><span style="font-family: courier new,monospace;">> 1110 = 432 = 9</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">> 1111 = 4321 = 10</span><br style="font-family: courier new,monospace;"><span style="font-family: courier new,monospace;">> 1120 = 4322 = 11</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">> 1210 = 4332 = 12</span><br style="font-family: courier new,monospace;"><span style="font-family: courier new,monospace;">> 2110 = 4432 = 13</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">> 11110 = 5432 = 14</span><br style="font-family: courier new,monospace;"><span style="font-family: courier new,monospace;">> 11111 = 54321 = 15</span><br style="font-family: courier new,monospace;">
<br>
In Andrew's representation, one way to determine it is to find t(n) as
the biggest triangular number less than n. Then, write the digits
corresponding to t(n) [for instance, if t(n) is 10, write 4321].<br>
<br>
Then, take k = n - t(n), and add 1 to each of the last k digits that
you have. For instance, if n is 13, write 4321 and add 1 to each
of the last three digits, 4432.<br>
<br>
In my representation, if n is triangular then write 111...1.<br>
Otherwise, with k defined as in the previous paragraph, write that same
string of 1s (corresponding to t(n)), replace the last digit with a 0,
and replace the k+1st digit from the right with a 2.<br>
<br>
I kinda like the "place value" of my representation, though Andrew's algorithm is a fun way to obtain his representation.<br>
<br>
Oh, and should the offset be 1, or should we insert a 0 at the beginning?<br>
<br>
--Joshua Zucker<br>
<br>
PS: more terms for Andrew's notation: Hey, wait, it breaks down
at 54 or so, when you start wanting to write 10s in there! <br>
1<br>
2<br>
21<br>
22<br>
32<br>
321<br>
322<br>
332<br>
432<br>
4321<br>
4322<br>
4332<br>
4432<br>
5432<br>
54321<br>
54322<br>
54332<br>
54432<br>
55432<br>
65432<br>
654321<br>
654322<br>
654332<br>
654432<br>
655432<br>
665432<br>
765432<br>
7654321<br>
7654322<br>
7654332<br>
7654432<br>
7655432<br>
7665432<br>
7765432<br>
8765432<br>
87654321<br>
87654322<br>
87654332<br>
87654432<br>
87655432<br>
87665432<br>
87765432<br>
88765432<br>
98765432<br>
987654321<br>
987654322<br>
987654332<br>
987654432<br>
987655432<br>
987665432<br>
987765432<br>
988765432<br>
998765432<br>
(in the subsequent two terms, the first "digit" is 10)<br>
1098765432<br>
10987654321<br>
<br>
and terms for my notation for the same sequence:<br>
1<br>
10<br>
11<br>
20<br>
110<br>
111<br>
120<br>
210<br>
1110<br>
1111<br>
1120<br>
1210<br>
2110<br>
11110<br>
11111<br>
11120<br>
11210<br>
12110<br>
21110<br>
111110<br>
111111<br>
111120<br>
111210<br>
112110<br>
121110<br>
211110<br>
1111110<br>
1111111<br>
1111120<br>
1111210<br>
1112110<br>
1121110<br>
1211110<br>
2111110<br>
11111110<br>
11111111<br>
11111120<br>
11111210<br>
11112110<br>
11121110<br>
11211110<br>
12111110<br>
21111110<br>
111111110<br>
111111111<br>
111111120<br>
111111210<br>
111112110<br>
111121110<br>
111211110<br>
112111110<br>
121111110<br>
211111110<br>
1111111110<br>
1111111111<br>
1111111120<br style="font-family: courier new,monospace;">