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Consecutive Triangles (Posted on 2009-11-29) Difficulty: 4 of 5
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 possible1) how such sets of values are distributed across the number system.

1. This can be explained in terms of a single variable expression.

See The Solution Submitted by brianjn    
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re(2): an observation | Comment 6 of 8 |
(In reply to re: an observation by brianjn)

I think the use of SQRT(3n2 + 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 2009-11-30 11:35:10
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