论文标题
三角类型的属性和平衡，卢卡斯平衡，同级和卢卡斯的平衡数字的均等数字
Trigonometric-type properties and the parity of balancing, Lucas-balancing, cobalancing and Lucas-cobalancing numbers
论文作者
论文摘要
平衡数字$ n $最初定义为Diophantine方程的解决方案$ 1+2+\ cdots+（n-1）=（n+1）+\ cdots+（n+r）$，其中$ r $称为平衡器，对应于平衡数字$ n $。通过稍作修改，$ n $是与co级$ r $的sto级数字，如果$ 1+2+\ cdots+n =（n+1）+\ cdots+（n+r）$。令$ b_n $表示$ n^{th} $平衡号码和$ b_n $表示$ n^{th} $ co平衡号码。然后$ 8b_n^2+1 $和$ 8b_n^2+8b_n+1 $是完美的正方形。 $ n^{th} $ lucas平衡号$ c_n $和$ n^{th} $ lucas-cobalancing number $ c_n $是$ 8B_N^2+1 $和$ 8B_N^2+8b_n+1 $的正源。在本文中，我们建立了一些三角类型的身份，以及有关平衡，so谐，卢卡斯平衡和卢卡斯 - 赛车数字的均衡性的一些算术特性。
Balancing numbers $n$ are originally defined as the solution of the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+\cdots+(n+r)$, where $r$ is called the balancer corresponding to the balancing number $n$. By slightly modifying, $n$ is the cobalancing number with the cobalancer $r$ if $1+2+\cdots+n=(n+1)+\cdots+(n+r)$. Let $B_n$ denote the $n^{th}$ balancing number and $b_n$ denote the $n^{th}$ cobalancing number. Then $8B_n^2+1$ and $8b_n^2+8b_n+1$ are perfect squares. The $n^{th}$ Lucas-balancing number $C_n$ and the $n^{th}$ Lucas-cobalancing number $c_n$ are the positive roots of $8B_n^2+1$ and $8b_n^2+8b_n+1$, respectively. In this paper, we establish some trigonometric-type identities and some arithmetic properties concerning the parity of balancing, cobalancing, Lucas-balancing and Lucas-cobalancing numbers.
