$$\\sum_{i=0}^{2^{n}}{lowbit(i)} $$
求 \[\sum_{i=0}^{2^{n}}{lowbit(i)} \] \((n≤10^{18})\) 答案对998244353取模 lowbit 首先数组数组里经常出现的lowit函数 对于lowbit 给定非负整数n。记lowbit(x)为x的二进制表示下最低位的1所对应的值 比如 lowbit(1),1的二进制位1,最低为对应的是1 lowbit(3),3的二进制为11,最低位对应的是1 lowbit(4),4的二进制为100,最低位对应的是4 一般来说2^n的lowbit为本身,奇数的lowbit为1 比如n=2 \(\sum_{i=0}^{2^{n}}{lowbit(i)}\\ = \sum_{i=0}^{4}{lowbit(i)}\\ = lowbit(0) + lowbit(1)+ lowbit(2)+ lowbit(3)+ lowbit(4)\) 而对于 \(lowbit(0)=0,lowbit(2^{n})=2^{n}\) 而对于 \(2^{n}有2^{n-1}个奇数\) 所以式子变成了求 \(lowbit(2) + lowbit(4) +lowbit(6).....+lowbit(2^{n-1})+2^{n}+2^{n-1}\) \(令x=lowbit(2) + lowbit(4) +lowbit(6).....+lowbit(2^{n-1})\) 打表