There are 40 ways to make sums of three distinct positive integers total 25. (1+2+22 is such a sum, but 1+12+12 and 1+2+3+19 are not.)

How many different ways can three distinct positive integers sum to 1000?

(In reply to re(2): Curious second differences by Old Original Oskar!)

I have verified algebraically that your formula is completely equivalent to the ones I gave in my previous posting.

Here is a fast QBASIC program that numerically verifies your formula for any long integer:

DEFLNG A-Z
PRINT "Enter N and press ENTER": INPUT N: PRINT " N ="; N
total = 0
FOR a = 1 TO (N \ 3) - 1
b = a + 1
c = N - a - b
        WHILE c > b
                total = total + 1
                b = b + 1
                c = N - a - b
        WEND
NEXT
K = N \ 6
L = N - 6 * K
M = 0: IF L = 0 THEN M = 1
PRINT total, 3 * K * K + (L - 3) * K + M

Edited on July 24, 2004, 1:44 am

Posted by Richard on 2004-07-23 21:00:08