Sumy z sześciokątną piramidą z kul
: 22 paź 2024, o 05:20
\(\displaystyle{ a _{n} - \hbox{ liczba kul w piramidzie sześciokątnej, n-warstwowej} }\)
\(\displaystyle{ a _{n} = 2 ^{-4} \left( \left( n+1\right) ^{4} - n ^{4} -\left( -1\right) ^{n} \right) }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{a _{n} }{k ^{n} } = \frac{k ^{4} + k ^{3} + k ^{2} }{\left( k ^{3} - k ^{2} - k + 1 \right) \left( k - 1\right) ^{2} } , k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{\left( -1\right) ^{n+1} a _{n} }{k ^{n} } = \frac{k ^{4} - k ^{3} + k ^{2} }{\left( k ^{3} + k ^{2} - k - 1 \right) \left( k + 1\right) ^{2} } , k \ge 2 }\)
\(\displaystyle{ a _{n} = 2 ^{-4} \left( \left( n+1\right) ^{4} - n ^{4} -\left( -1\right) ^{n} \right) }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{a _{n} }{k ^{n} } = \frac{k ^{4} + k ^{3} + k ^{2} }{\left( k ^{3} - k ^{2} - k + 1 \right) \left( k - 1\right) ^{2} } , k \ge 2 }\)
\(\displaystyle{ \sum_{n=1}^{ \infty } \frac{\left( -1\right) ^{n+1} a _{n} }{k ^{n} } = \frac{k ^{4} - k ^{3} + k ^{2} }{\left( k ^{3} + k ^{2} - k - 1 \right) \left( k + 1\right) ^{2} } , k \ge 2 }\)