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40 lines
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40 lines
2.0 KiB
HTML
<p>You are given an integer <code>n</code>. You roll a fair 6-sided dice <code>n</code> times. Determine the total number of <strong>distinct</strong> sequences of rolls possible such that the following conditions are satisfied:</p>
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<ol>
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<li>The <strong>greatest common divisor</strong> of any <strong>adjacent</strong> values in the sequence is equal to <code>1</code>.</li>
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<li>There is <strong>at least</strong> a gap of <code>2</code> rolls between <strong>equal</strong> valued rolls. More formally, if the value of the <code>i<sup>th</sup></code> roll is <strong>equal</strong> to the value of the <code>j<sup>th</sup></code> roll, then <code>abs(i - j) > 2</code>.</li>
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</ol>
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<p>Return <em>the<strong> total number</strong> of distinct sequences possible</em>. Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>
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<p>Two sequences are considered distinct if at least one element is different.</p>
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<p> </p>
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<p><strong>Example 1:</strong></p>
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<pre>
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<strong>Input:</strong> n = 4
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<strong>Output:</strong> 184
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<strong>Explanation:</strong> Some of the possible sequences are (1, 2, 3, 4), (6, 1, 2, 3), (1, 2, 3, 1), etc.
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Some invalid sequences are (1, 2, 1, 3), (1, 2, 3, 6).
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(1, 2, 1, 3) is invalid since the first and third roll have an equal value and abs(1 - 3) = 2 (i and j are 1-indexed).
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(1, 2, 3, 6) is invalid since the greatest common divisor of 3 and 6 = 3.
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There are a total of 184 distinct sequences possible, so we return 184.</pre>
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<p><strong>Example 2:</strong></p>
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<pre>
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<strong>Input:</strong> n = 2
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<strong>Output:</strong> 22
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<strong>Explanation:</strong> Some of the possible sequences are (1, 2), (2, 1), (3, 2).
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Some invalid sequences are (3, 6), (2, 4) since the greatest common divisor is not equal to 1.
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There are a total of 22 distinct sequences possible, so we return 22.
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</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>1 <= n <= 10<sup>4</sup></code></li>
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</ul>
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