There's a nice, well-known theory of which integers can be expressed as the sum of two squares. What can be said about numbers that are sums of two triangles, i.e. 1 or 3 or 6 or 10 or 15 ... + 1 or 3 or 6 or 10 or 15 ...? Bill
It is sequence A051533 but no characterization is included. Emeric On Sat, 18 Dec 2004, William Thurston wrote:
There's a nice, well-known theory of which integers can be expressed as the sum of two squares. What can be said about numbers that are sums of two triangles, i.e. 1 or 3 or 6 or 10 or 15 ... + 1 or 3 or 6 or 10 or 15 ...?
Bill
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
At 11:03 PM 12/18/2004, William Thurston wrote:
There's a nice, well-known theory of which integers can be expressed as the sum of two squares. What can be said about numbers that are sums of two triangles, i.e. 1 or 3 or 6 or 10 or 15 ... + 1 or 3 or 6 or 10 or 15 ...?
An integer N is the sum of two triangles if and only if 4N + 1 is the sum of two squares. Proof: If N is the sum of two triangles, say N = a(a + 1)/2 + b(b + 1)/2, then 4N + 1 = 2a^2 + 2a + 2b^2 + 2b + 1 = (a + b + 1)^2 + (a - b)^2. Conversely, if 4N + 1 is the sum of two squares, say 4N + 1 = u^2 + v^2, then u and v necessarily have opposite parity, so they can be written as a + b + 1 and a - b for some integers a,b; hence we again have N = a(a + 1)/2 + b(b + 1)/2. -- Fred W. Helenius <fredh@ix.netcom.com>
William Thurston asks if there is a characterization of numbers that can be expressed as the sum of two triangles (A020756), in the spirit of the characterization of a sum of two squares (A001481). Emeric Deutsch earlier submitted that the sequence in question was A051533, the sums of two positive triangular numbers. This is a much more difficult beastie to characterize, and I don't think it was intended. Anyway, Fred Helenius correctly characterized A020756 as the numbers n such that 4n+1 is in A001481. Thus characterizations of A001481 transform to characterizations of A001481. Specifically %C A001481 Closed over multiplication. [Not in OEIS] %C A020756 4n+1 = odd square * product of distinct primes of form 4k+1. %C A020756 Closed over f(x, y) = 4xy + x + y. Another nice property noted on A020756 is that the sums of two triangles are precisely the sums of a square and a pronic.
Thanks for the replies --- this is all helpful. Let me give the context, why I'm interested. I have a friend who's working on plans for a "geometry playground" traveling science exhibit, and part of this project is coordinated with an actual playground manufacturer who is hoping for spinoffs of some of the ideas for larger-scale distribution. One of the ideas is to make a "multiplication table" climber. The key criterion (for real playgrounds) is for it to be fun to play on---the people who buy these things aren't primarily interested in any explicit educational value. I played around a little and came up with a design obtained from the parabolic hyperboloid z = x * y by distorting coordinates in domain on each axis to emphasize integral points. Take a look at the pictures (done with MMA) at http://homepage.mac.com/wpthurston/PhotoAlbum1.html. You need to click on the slide show to appreciate what's going on. I think kids will spontaneously enjoy following the levels, which widen out at factorizations; the levels that never widen out are at prime heights. If something like this gets built, it will probably be about 8' square, as a way to get up to a deck with slides etc; it could be intriguing to kids because of the patterns, e.g. trying to go around always staying on the same level. Numbers would be engraved on the squares showing the heights, perhaps accompanied by an n x m array of dots. The other two pictures are graphs showing z = x^2 + y^2 and z = x(x-1)/2 + y(y-1)/2. The latter really goes with the multiplication table graph better, but it's harder to think of good designs. For the sum of two squares, one could put two square arrays of dots along with a square of the appropriate area with corners on lattice points. For the sum of triangular numbers, one could always put two equilateral triangles of dots, but it would be nice to see an illustration as the area of something natural. Maybe something can be worked out from the 4n+1 - sum of two squares idea. I had hoped there would be a way to see it in the triangular lattice---maybe there is, I just don't see it yet. Bill
participants (5)
-
David Wilson -
Emeric Deutsch -
Fred W. Helenius -
William Thurston -
William Thurston