# Friendly Numbers

Level pending

A positive integer $$n$$ is called friendly if there exist two positive integers $$j$$ and $$k$$ such that:

• $$j+k= n$$

• There exist two sequences $$\{a_i\}_{i=0}^{\infty}$$ and $$\{b_i\}_{i=0}^{\infty}$$ of integers (not necessarily positive) satisfying the following conditions. $\begin{cases} a_i=0 & \text{for all } i>j \\ b_i=0 &\text{for all } i>k \\ \displaystyle \sum_{i=0}^{v} a_i b_{v-i} = 0 & \text{for all } v \in \{1, 2, \cdots , n-1 \} \\ a_0b_0= a_jb_k= 1 \\ \end{cases}$

Find the number of positive integers $$\leq 1000$$ which are not friendly.

Details and assumptions

• As an explicit example, when $$n=3$$ and $$j=1, k=2$$, the last condition reads: $\begin{cases} a_i=0 & \text{for all } i>1 \\ b_i=0 & \text{for all } i>2 \\ a_0b_0= a_1b_2= 1 \\ a_0b_1 + a_1b_0= 0 \\ a_0b_2 + a_1b_1 + a_2b_0= 0 \end{cases}$ It isn't hard to verify that the sequences $$\{a_0, a_1, a_2, a_3, \cdots \} = \{1, 1, 0, 0, \cdots \}$$ and $$\{b_0, b_1, b_2, b_3, \cdots \} = \{1, -1, 1, 0, 0, \cdots \}$$ satisfy all the conditions above, so $$3$$ is a friendly number.

• By definition, $$1$$ and $$2$$ are unfriendly numbers.

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