Concatenations

Take the numbers 1, 2, and 3 and concatenate them as 123, or one hundred twenty-three.  Then, reverse the order and concatenate them as 321, or three hundred twenty-one.  Add them and you get 444.  This is the third element of a sequence that begins:

• A(1) = 1 + 1 = 2
• A(2) = 12 + 21 = 33
• A(3) = 123 + 321 = 444
• A(4) = 1234 + 4321 = 5555, etc.

It seems pretty straightforward until the ninth element, when the carries start to complicate things.  The first 20 steps can be illustrated by using disks of color.  Here, each row represents a step in the sequence, with the first number (2) being represented by the bottom row (the single gray dot on the right).  The next row up shows 33 with two lighter gray dots.  In general, the shade of gray represents the digit, from black for 0 to white for 9.

The above image shows that a change from rows of constant color to other structures happens around the 10th step in the sequence.  Other interesting changes happen around the 100th step, as shown below.  The rows represent steps 91 (bottom) through 190 (top), and the first 200 digits (from right to left).

Around the 1000th step, things change again.  This image (below) shows steps 926 (bottom) through 1125 (top), and the first 400 digits of the sums.

Notice the black horizontal line across the left half of the image, about one-quarter of the way up from the bottom.  This happens at step 962.  Black means a digit of 0, so this shows the beginning of a line of a few thousand zeroes.  Or, 12345…960961962 + 962961960…54321 = a 2,779-digit number with a run of 2,402 zeroes in the middle of it.  I’m still looking for other long runs of zeroes, but haven’t found any yet.

Different, but similar images arise when subtracting the numbers rather than adding.  The image at the top of this page combines both ideas; the inner dots represent the digits of the differences between the concatenated numbers and the backgrounds of the square cells correspond the the digits of the sums.