Rather Unique!

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1 , 2 , 9 , 44 , 265 , ? \large 1, \ 2, \ 9, \ 44, \ 265, \ ?

What is the next number in the above sequence?


The answer is 1854.

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3 solutions

Mehul Arora
May 17, 2016

The terms of the pattern are the number of Derangements of Natural numbers

Thus the next term is 7 ! × ( 1 2 1 6 + 1 24 1 120 + 1 720 1 5040 ) = 185 4 7! \times \left( \dfrac {1}{2} - \dfrac {1}{6} + \dfrac {1}{24} - \dfrac {1}{120} + \dfrac {1}{720} - \dfrac {1}{5040} \right) = 1854_{\square}

This problem has been posted several times... @Geoff Pilling

Mateo Matijasevick - 5 years ago

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Yup, its similar to this one. :^)

Geoff Pilling - 5 years ago

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Well, I guess it's not unique. It's still a great problem though!

Seth-Riley Adams - 5 years ago

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@Seth-Riley Adams Yup... Definitely! :)

Geoff Pilling - 5 years ago

Also known as subfactorials.

Seth-Riley Adams - 5 years ago

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Yep! Subfactorials it is.

Mehul Arora - 5 years ago

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Nice problem! It's... unique! :)

Seth-Riley Adams - 5 years ago

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@Seth-Riley Adams Ahahah Thanks

Mehul Arora - 5 years ago

Cool !!! Nice question! (Ps. I got it !! Haha)

abc xyz - 5 years ago

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Haha great :)

Mehul Arora - 5 years ago
Ashish Menon
May 17, 2016

( 1 1 ×3)-1 => ( 2 2 ×4)+1 => ( 9 9 ×5)-1 => ( 44 44 => ( 44 44 ×6)+1 => ( 265 265 ×7)-1 = 1854 \boxed{1854} .

Marc Tailheuret
May 22, 2016

Hi. Let's suppose that a 1 = 1 , a 2 = 2 a_{1}=1, a_{2}=2 we got a n = n ( a n 1 + a n 2 ) a_{n}=n*(a_{n-1}+a_{n-2}) that suits, thus a 6 = 6 ( 44 + 265 ) = 1854 a_{6}=6*(44+265)=1854

Moderator note:

Good job observing the recurrence relation, Marc. The term, a n a_n , denotes the number of derangements of n n distinct objects. Could you prove this recurrence relation using a combinatorial approach?

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