| | Ivars Peterson's MathTrek |
November 18, 1996
Consider, for example, the sequence of counting numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ....
Now, take out every second number, leaving: 1 3 5 7 9 11 13 15 ...; form the cumulative totals of these numbers: 1 (1 + 3) (4 + 5) (9 + 7) (16 + 9) (25 + 11) (36 + 13) (49 + 15) ...; and out pops the sequence of consecutive squares: 1 4 9 16 25 36 49 64...!
This seemingly magical transformation of one sequence into another was first discovered and explored by mathematician Alfred Moessner in the early 1950s. He and others found a host of such relationships between different number sequences.
Again, starting with the sequence of counting numbers, suppose you take out every third number, adding up what's left to get cumulative totals, then remove every second number in the new list, and total the remaining numbers. What do you end up with?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
1 2 4 5 7 8 10 11 13 14 ...
1 3 7 12 19 27 37 48 61 75 ...
1 7 19 37 61 ...
1 8 27 64 125 ...
The sequence of cubes!
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 16 81 256 .... In general, taking out the nth number to start with gives a sequence of nth powers in the end.
What happens if you take out the so-called triangular numbers: 1 (1 + 2) (1 + 2 + 3) (1 + 2 + 3 + 4) ... (1 + 2 + 3 + 4 + ... n), and as before, calculate cumulative totals, then take out the appropriate numbers from the new list, and so on, as above?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
2 4 5 7 8 9 11 12 13 14 ...
2 6 11 18 26 35 46 58 71 85 ...
6 18 26 46 58 71 ...
6 24 50 96 154 225 ...
24 96 154 ...
24 120 274 ...
120 ...
Notice that the numbers down the left hand side, 1 2 6 24 120 are the factorial numbers: 1 (1 x 2) (1 x 2 x 3) (1 x 2 x 3 x 4) (1 x 2 x 3 x 4 x 5), or in general, (1 x 2 x 3 x ... x n). Somehow, the recipe turns addition into multiplication.
I first came across these surprising sequence transformations when Richard Guy described them at a meeting on recreational mathematics held in 1986 at the University of Calgary. Now this material - and much, much more - is included in a fascinating new book by Guy and John Conway called The Book of Numbers. If you want to stretch your mind from the integers to the surreals, this is the book to read!
Guy, Richard K., and Robert E. Woodrow, eds. 1994. The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History. Washington, D.C.: Mathematical Association of America.
Peterson, Ivars. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York, W.H. Freeman.
Sloane, N.J.A., and S. Plouffe. 1995. The Encyclopedia of Integer Sequences. New York: Academic Press.