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When I run up a staircase, my stride can carry me up $1$, $2$, or $3$ steps at a time. In how many ways can I run up a $9$-step staircase (given that my last stride lands me on the $9^{\text{th}}$ step)?

I can't do constructive and that's all I know.

Caleb Stanford
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Yuna Kun
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    I think this might get you started: http://math.stackexchange.com/questions/789804/how-many-distinct-ways-to-climb-stairs-in-1-or-2-steps-at-a-time?noredirect=1&lq=1 – turkeyhundt Sep 19 '16 at 23:01
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    Do you count $3+1+2+3$ and $1+3+3+2$ as different ways for climbing the stairs? – Jack D'Aurizio Sep 19 '16 at 23:01
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    Hint: try to set up a recursion. Let $S_n$ be the number of ways to reach step $n$ and note that if you made it to step $n$ the next to last step must have been one of $n-1, n-2, n-3$. – lulu Sep 19 '16 at 23:01
  • I'm going to call it a duplicate of this: http://math.stackexchange.com/questions/868024/number-of-ways-of-walking-up-stairs-and-recurrence-relation?rq=1. The difference between $6$ and $9$ is very marginal. – Caleb Stanford Sep 19 '16 at 23:06

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Let $W_n$ the number of ways for climbing a stair with length $n$ with steps having size $\in\{1,2,3\}$.
We clearly have $$ W_1 = 1,\qquad W_2= 2,\qquad W_3=4 $$ and since the last step may only be $+1,+2$ or $+3$, for any $n\geq 4$ we have: $$ W_{n} = W_{n-1}+W_{n-2}+W_{n-3}.$$ Now see tribonacci numbers. We have $W_9=\color{red}{149}$.

Jack D'Aurizio
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