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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? –  Zev Chonoles Dec 25 '12 at 6:11
    
@ZevChonoles : perhaps you can elaborate? –  ainunnajib Dec 25 '12 at 6:17
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What is the probability measure you are putting on the set of positive integers? –  Zev Chonoles Dec 25 '12 at 6:22
    
@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]? –  ainunnajib Dec 25 '12 at 6:30
    
Is it possible to generate many of such random sequence by a computer program and see how often they satisfy the properties you describe? –  Fitri Dec 25 '12 at 7:20
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1 Answer

The integer wont exceed 3300 If Sn = S even Fn; Sn = S even n + S odd n S even F n = S even n + T even n

Then S even n + S odd n = S even n + T even n S odd n = T even n T even n is 3306 S odd n = 3306 The sum of 57 random positive integer is 3306. Is it safe to use combination of 3306 and 57? As for the other one: Tn = S even Fn Tn = T even n + T odd n S even Fn = S even n + T even n Thus T even n + T odd n = S even n + T even n T odd n = S even n T even n is 3249 S even n = 3249

Combination of 3249 and 57

What I still wonder is, how to determine all possibility used in this to divide them

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how exactly does this answer the question? –  Fitri Dec 25 '12 at 13:51
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