Understanding LOST sequence of numbers

Anyone who’s a fan of LOST is familiar with a sequence of numbers: 4 8 15 16 23 42. These are the numbers that almost connect all strange things in the series. During the second season of LOST it was revealed that the numbers are the code that must be entered into the computer located inside The Swan every 108 minutes (by the way 4 + 8 + 15 + 16 + 23 + 42 = 108). Entering the numbers resets the countdown timer to 108 minutes. If an individual does not push the button in time, the numbers flip to a series of glyphs . While the numeric sequence is flipping into these icons, an individual can still finish typing in the numbers, press execute, and return the counter to 108 minutes. Where do these numbers come from? In mathematics, there’s something known as the Shaw-Basho Polynomial. It’s an equation that reads:

There’s also something known as Pascal’s triangle. It is a geometric arrangement of the binomial coefficients in a triangle. In simple terms, Pascal’s triangle can be constructed in the following manner. On the first row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left and the number directly above and to the right to find the new value. If either the one to the right or left is not present, substitute a zero in its place. For example, the numbers 1 and 3 in the fourth row are added to produce 4 in the fifth row. Doing this to fill 11 rows gives you the following:

Now, take into account only the first six rows of Pascal’s triangle. That is:

1
1   1
1   2   1
1   3   3   1
1   4   6   4   1
1   5  10  10   5   1

Next, alternate signs for the above triangle, leaving it as follows (bolded numbers are those which sign changed):

1
-1   1
1   -2   1
-1   3   3   1
1   -4   6   -4   1
-1   5  -10  10   -5   1

Go back to the Shaw-Basho polynomial and fill a vector with s(0), s(1), s(2), s(3), s(4), s(5), where s(x) is the result of calculating the polynomial replacing the value of x. When you fill in a matrix with above changed version of Pascal’s triangle (replacing with 0 missing columns), and then multiply with this vector, you’ll get something interesting:

That is, the result is a vector consisting of LOST numbers: 4 8 15 16 23 42.

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