相鄰兩項的平均是完全平方

hbghlyj

證明存在具有以下性質的無限完全平方数序列：
(i) 任意相鄰兩項的算術平均值是完全平方，
(ii) 任意相鄰兩項互質，
(iii) 序列嚴格遞增。

hbghlyj

aops

Define the sequence $ ( s_{n})$ by $ s_{n} =\left(( 2n+1)^{2} -2\right)^{2} \ \ ( n\geq 0)$.

It is clear that $ ( s_{n})$ is strictly increasing.

Since $ s_{n} +s_{n+1} =2\left( 4n^{2} +8n+5\right)^{2}$, $ ( s_{n} +s_{n+1}) /2$ is a perfect square.

We can show that $ \gcd( s_{n} ,s_{n+1}) =1$ as follows :
It is sufficient to show that $ \gcd\left(( 2n+1)^{2} -2,( 2n+3)^{2} -2\right) =1$.
$ \gcd\left(( 2n+1)^{2} -2,( 2n+3)^{2} -( 2n+1)^{2}\right) =\gcd\left(( 2n+1)^{2} -2,8( n+1)\right)$
$ =\gcd\left(( 2( n+1) -1)^{2} -2,n+1\right) =\gcd\left( 1^{2} ,n+1\right) =1$.