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It looks simple but can you explain it?

How would you describe the sequence of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, ...?

Nota Bene: You're not allowed use a computer program or code to describe it.

Note by Hobart Pao
11 months ago

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Hint: Apply the general formula for \(C_{10} \) for Champernowne constant. Then 10 raise to the power of respective digits, then mod 10 answer. The full working is very long and ugly. Pi Han Goh · 11 months ago

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@Pi Han Goh So,

\[T_n = \lfloor 10^{n} \times C_{10} \rfloor \pmod{10}\] Sharky Kesa · 11 months ago

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@Sharky Kesa AKA

\[\large T_k = \left \lfloor 10^k \times \displaystyle \sum_{m=0}^{\infty} \sum_{n=10^{m-1}}^{10^m - 1} \dfrac {n}{10^{m(n - 10^{m - 1} + 1) + 9 \sum_{l=1}^{m-1} 10^{l-1} l}} \right \rfloor \pmod{10}\] Sharky Kesa · 11 months ago

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@Pi Han Goh Isn't \(C_{10} = 0.123456789101112131415 \ldots\)? I don't see why there'd be a problem with the relation. Sharky Kesa · 11 months ago

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@Sharky Kesa Oh wait. You're right. I was thinking of something completely different. Thanks! =D

But since OP wanted the full formula, you might want to type out that nasty double summation in full. HAHA! Pi Han Goh · 11 months ago

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@Pi Han Goh Written out the LaTeX hell. Had to put it in large so it could be read. Sharky Kesa · 11 months ago

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@Sharky Kesa My eyes is in pain just by staring at it! Pi Han Goh · 11 months ago

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Thanks guys! I love what you guys did! I thought it'd be simpler because this was on an entrance exam for a specialized middle school. Hobart Pao · 11 months ago

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