Powerful Sequences

Number sequences suggest all sorts of intriguing puzzles and patterns.

Consider, for example, the sequence of counting numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …

Take out every second number, leaving just the odd integers: 1, 3, 5, 7, 9, 11, 13, 15, …

Then, form the cumulative totals of the odd numbers, as follows:

1, (1 + 3) = 4, (4 + 5) = 9, (9 + 7) = 16, (16 + 9) = 25, (25 + 11) = 36, (36 + 13) = 49, (49 + 15) = 64, …

Out pops the sequence of consecutive squares: 1 = 1², 4 = 2², 9 = 3², 16 = 4², 25 = 5², 36 = 6², 49 = 7², 64 = 8², and so on.

This seemingly magical transformation of one sequence into another was studied by German mathematician Alfred Moessner in the early 1950s. He found a host of such relationships between different number sequences.

Again starting with the sequence of counting numbers, suppose you take away every third number (multiples of 3): 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, …

Then, add up what's left to get cumulative totals:

1, (1 + 2) = 3, (3 + 4) = 7, (7 + 5) = 12, (12 + 7) = 19, (19 + 8) = 27, (27 + 10) = 37, (37 + 11) = 48, (48 + 13) = 61, (61 + 14) = 75, (75 + 16) = 91, …

You end up with the following sequence: 1, 3, 7, 12, 19, 27, 37, 48, 61, 75, 91, …

Remove every second number in the new list: 1, 7, 19, 37, 61, 91, …

Calculate the cumulative totals of the remaining numbers. What do you end up with?

1, (1 + 7) = 8, (8 + 19) = 27, (27 + 37) = 64, (64 + 91) = 125, (125 + 91) = 216, …

You get the sequence of cubes: 1 = 1³, 8 = 2³, 27 = 3³, 64 = 4³, 125 = 5³, 216 = 6³, and so on.

If you go through the same procedure again, this time striking out every fourth number at the start, the result should now come as no surprise. You end up with the sequence of fourth powers: 1 = 1⁴, 16 = 2⁴, 81 = 3⁴, 256 = 4⁴, …

In general, taking out the nth number and following the appropriate procedure gives a sequence of nth powers. What happens if you take out the so-called triangular numbers:

1, (1 + 2) = 3, (1 + 2 + 3) = 6, (1 + 2 + 3 + 4) = 10, (1 + 2 + 3 + 4 + 5) = 15, …, (1 + 2 + 3 + 4 + . . . + n)

and as before, calculate cumulative totals, then take out the first, third, sixth, tenth, fifteenth, and so on numbers from the new list, then continue on, as above?

Notice that the unique numbers down the left-hand side are the factorial numbers: 1, (1 x 2) = 2, (1 x 2 x 3) = 6, (1 x 2 x 3 x 4) = 24, (1 x 2 x 3 x 4 x 5) = 120, or in general, (1 x 2 x 3 x ... x n). Somehow, the recipe turns addition into multiplication. For more information about a related sequence, see Moessner’s factorial triangle at the On-Line Encyclopedia of Integer Sequences.

I first came across these surprising sequence transformations when Richard K. Guy described them at a meeting on recreational mathematics held in 1986 at the University of Calgary. This material—and much, much more—is included in a fascinating book by Guy and John H. Conway titled The Book of Numbers (Springer, 1996).

If you want to stretch your mind from the integers to the surreal, this is the book to read!

Originally posted November 18, 1996