ファウルハーバーの公式
S 0 ( n ) = n S 1 ( n ) = n 2 + n 2 = 1 2 n ( n + 1 ) S 2 ( n ) = 2 n 3 + 3 n 2 + n 6 = 1 6 n ( n + 1 ) ( 2 n + 1 ) S 3 ( n ) = n 4 + 2 n 3 + n 2 4 = 1 4 n 2 ( n + 1 ) 2 S 4 ( n ) = 6 n 5 + 15 n 4 + 10 n 3 − n 30 = 1 30 n ( n + 1 ) ( 2 n + 1 ) ( 3 n 2 + 3 n − 1 ) S 5 ( n ) = 2 n 6 + 6 n 5 + 5 n 4 − n 2 12 = 1 12 n 2 ( n + 1 ) 2 ( 2 n 2 + 2 n − 1 ) S 6 ( n ) = 6 n 7 + 21 n 6 + 21 n 5 − 7 n 3 + n 42 = 1 42 n ( n + 1 ) ( 2 n + 1 ) ( 3 n 4 + 6 n 3 − 3 n + 1 ) {\displaystyle {\begin{aligned}S_{0}(n)&=n\\S_{1}(n)&={\frac {n^{2}+n}{2}}={\frac {1}{2}}n(n+1)\\S_{2}(n)&={\frac {2n^{3}+3n^{2}+n}{6}}={\frac {1}{6}}n(n+1)(2n+1)\\S_{3}(n)&={\frac {n^{4}+2n^{3}+n^{2}}{4}}={\frac {1}{4}}n^{2}(n+1)^{2}\\S_{4}(n)&={\frac {6n^{5}+15n^{4}+10n^{3}-n}{30}}={\frac {1}{30}}n(n+1)(2n+1)(3n^{2}+3n-1)\\S_{5}(n)&={\frac {2n^{6}+6n^{5}+5n^{4}-n^{2}}{12}}={\frac {1}{12}}n^{2}(n+1)^{2}(2n^{2}+2n-1)\\S_{6}(n)&={\frac {6n^{7}+21n^{6}+21n^{5}-7n^{3}+n}{42}}={\frac {1}{42}}n(n+1)(2n+1)(3n^{4}+6n^{3}-3n+1)\end{aligned}}}