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Say we have a sequence of $n$ positive integers, we can assume they're randomly chosen, let's call it $U_{n}$.

Let $S_{n}$ = sum of $U_{n}$ from $1$ to $n$.

Let $T_{n}$ = sum of $n$ from $1$ to $n$. In other words, $T_{n}$ is the triangle number $T_{n} = \frac{1}{2}n(n+1)$.

Define $F_{n} = U_{n} + n$.

(note that $SF_{n}$, sum of $F_{n}$, is equal to $S_{n} + T_{n}$)

Define two sets $EvenSet$ and $OddSet$ containing $F_{n}$ when it is even and odd respectively.

Let $SEvenF_{n}$ be the sum of numbers in $EvenSet$ and $SOddF_{n}$ be the sum of numbers in $OddSet$.

(note that $SEvenF_{n} + SOddF_{n} = SF_{n} = S_{n} + T_{n}$)

What is the probability of the two pairs ($S_{n}$ and $T_{n}$) and ($SEvenF_{n}$ and $SOddF_{n}$) are the same pairs?

In other words, either "$S_{n} = SEvenF_{n}$ thus $T_{n} = SOddF_{n}$" or "$S_{n} = SOddF_{n}$ thus $T_{n} = SEvenF_{n}$".

How to calculate this probability?

PS: I have a sample data with $n = 114$ in here http://goo.gl/k96FZ

PS2: Is there a way or an algorithm to generate such sequence?

PS3: @ZefChonoles has correctly pointed a concern about Probability Measure. Originally I intended this question to have no restriction on the positive integer set, that is to work on the infinite set of $\mathbb Z^{+}$.

But I believe you are free to impose certain restriction to this problem, as long as you can share insights on approaches in solving this problem.

For example, you can restrict $U_{n}$ and $F_{n}$ to be within $[1, 400]$ for the $n=114$ case. Or you can restrict them to be within $[1, Cn]$ with some constant $C$ for the general case. Or within $[1, Tn]$. Anything that helps, basically. Thanks!

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    "Randomly chosen" **in what sense**?2012-12-25
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    @ZevChonoles : perhaps you can elaborate?2012-12-25
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    What is the [probability measure](http://en.wikipedia.org/wiki/Probability_measure) you are putting on the set of positive integers?2012-12-25
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    @ZevChonoles : I see...indeed I have difficulty thinking through this problem due to the fact that positive integer is infinitely many. Is it necessary, or rather, will it help solving the problem if the numbers were restricted finitely, say only [1..1000]?2012-12-25
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    Is it possible to generate many of such random sequence by a computer program and see how often they satisfy the properties you describe?2012-12-25
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    @Fitri : Indeed Monte Carlo style programming came across my thought, while it does give certain insight (on how big/small is the probability), it will not give insight on how to reach there mathematically. Not to mention that we have to define and impose certain finite-restriction on the problem as computer simulation can only work within finite domain assumption. You are free to try and share the result by the way :-D2012-12-26

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