Neue Übersetzung für Problem 895

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Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns. On Gary's turn he chooses a gold coin and removes it from the game along with any other coins sitting on top. Sally does the same on her turn by removing a silver coin. The first player unable to make a move loses.

An arrangement is called <dfn>fair</dfn> if the person moving first, whether it be Gary or Sally, will lose the game if both play optimally.

An arrangement is called <dfn>balanced</dfn> if the number of gold and silver coins are equal.

Define $G(m)$ to be the number of fair and balanced arrangements consisting of three non-empty stacks, each not exceeding $m$ in size. Different orderings of the stacks are to be counted separately, so $G(2)=6$ due to the following six arrangements:

You are also given $G(5)=348$ and $G(20)=125825982708$.

Find $G(9898)$ giving your answer modulo $989898989$.

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<p>Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns. On Gary's turn he chooses a gold coin and removes it from the game along with any other coins sitting on top. Sally does the same on her turn by removing a silver coin. The first player unable to make a move loses.</p>​<p>An arrangement is called <dfn>fair</dfn> if the person moving first, whether it be Gary or Sally, will lose the game if both play optimally.</p>​<p>An arrangement is called <dfn>balanced</dfn> if the number of gold and silver coins are equal.</p>​<p>Define $G(m)$ to be the number of fair and balanced arrangements consisting of three non-empty stacks, each not exceeding $m$ in size. Different orderings of the stacks are to be counted separately, so $G(2)=6$ due to the following six arrangements:</p>​<div style="text-align:center;"><img src="resources/images/0895_G2.png?1714251811" alt="0895_G2.png"></div>​<p>You are also given $G(5)=348$ and $G(20)=125825982708$.</p>​<p>Find $G(9898)$ giving your answer modulo $989898989$.</p>

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