|
|
Wikipedia
初始值P(0)=P(1)=P(2)=1,
递推式P(n)=P(n-2)+P(n-3).
恒等式:\begin{align}P(k)&=P(k-2)+P(k-4)+P(k-8)\\
\sum_{k=0}^nP(2k+1)&=P(2n+4)-1
\end{align}
用DifferenceRootReduce验证:
PadovanNumber = DifferenceRoot[Function[{y, n}, {y[n + 3] - y[n + 1] - y[n] == 0, y[0] == 1, y[1] == 1, y[2] == 1}]];
DifferenceRootReduce[
PadovanNumber[k] == PadovanNumber[k - 2] + PadovanNumber[k - 4] + PadovanNumber[k - 8], k]
DifferenceRootReduce[
Sum[PadovanNumber[2 k + 1], {k, 0, n}] == PadovanNumber[2 n + 4] - 1, n]
输出
True
True |
|
