A, B, C and D are triangular numbers.
A, B and C are always consecutive while D is their sum.
Determine (and explain as best as possible^{1}) how such sets of values are distributed across the number system.
^{1.} This can be explained in terms of a single variable expression.
(In reply to
re: an observation by brianjn)
I think the use of SQRT(3n^{2} + 9n + 33/4)  (1/2) is a little confusing. Successive increments of n by 1 do not produce A082840, but rather, a sequence of numbers, some of which are integers and some are not. The n values that form x as integers form the sequence A082840. So for example, n=1 forms the integer x = 4, so the n=1 indicates the 1st triangular number serves as A. Values of n from 2 through 7 do not result in an integral x, so those are thrown out; n = 8 results in x = 16, an integer, so the 8th triangular number is the second A that works, and so on.
As the numbers get high, many x values will have to be calculated to determine which ones are integral so as to evaluate the validity of the given n used. Also, with the higher numbers it becomes harder to determine by computer which ones are integers and which ones are not; and by hand you wouldn't want to do so many calculations.

Posted by Charlie
on 20091130 11:35:10 