Posted by Soroban on September 11, 2002 at 13:31:36:
In Reply to: consecutive numbers posted by Jim on September 11, 2002 at 08:15:56:
: Sum of numbers from 1 to A = sum of numbers from A+1 to B.
: This is an example where A=14 and B=20:
: 1 + 2 ... + 14 = 105
: 15 + 16 ... + 20 = 105
: Is there a formula I could use to find more examples?
Yes, there is, Jim. But the derivation is intricate.
The sum of the numbers from 1 to A is A(+1)/2
The sum of the numbers from A+1 to B: B(B+1)/2 - A(A+1)/2
Setting them equal, we get a quadratic in A and B: 2A^2 + 2A - B(B+1) = 0
Solving for A: A = [sqrt(2B^2 + 2B + 1) - 1]/2
I am a rank beginner at Pellian Analysis, but I knew enough to seek
a recurrence relation and found one!
(A,B) = (0,0), (2,3), (14,20), (84,119),...
where: A(n) = 6*A(n-1) - A(n-2) + 2
and: B(n) = 6*B(n-1) - B(n-2) + 2, for n > 1.
The next two are (492,696) and (2870,4059).