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Omni-MATH / Assume n is a positive integer. Considers sequences a₀,a₁,…,a_(n) for which a_(i)∈{1,2,…,n} for all i and a_(n)=a₀. (a) Suppose n is odd. Find the number of such sequences if …

Problem

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problem

Assume

n

is a positive integer. Considers sequences

a_0, a_1, \ldots, a_n

for which

a_i \in \{1, 2, \ldots , n\}

for all

i

and

a_n = a_0

. (a) Suppose

n

is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all

i = 1, 2, \ldots, n

. (b) Suppose

n

is an odd prime. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$ for all

i = 1, 2, \ldots, n

Plain-text mathematical notation (without MathML)

Assume n is a positive integer. Considers sequences a₀,a₁,…,a_(n) for which a_(i)∈{1,2,…,n} for all i and a_(n)=a₀. 

(a) Suppose n is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all i=1,2,…,n.

(b) Suppose n is an odd prime. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$ for all i=1,2,…,n.

Original LaTeX notation

Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$. 

(a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$.

(b) Suppose $n$ is an odd prime. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$ for all $i = 1, 2, \ldots, n$.

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task_id: 9b1b5883-a0d8-5afc-ac00-732ab95a4ea7
task_key: test--9b1b5883-a0d8-5afc-ac00-732ab95a4ea7
task_revision_id: 4
config / split: / test

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