81 lines
5.8 KiB
HTML
81 lines
5.8 KiB
HTML
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<h2>Golomb's self-describing sequence</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 341</h3><span style="width:300px;color:#666;">Published on Sunday, 5th June 2011, 10:00 am; Solved by 508</span></div>
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<p>The <b>Golomb's self-describing sequence</b> {G(<var>n</var>)} is the only nondecreasing sequence of natural numbers such that <var>n</var> appears exactly G(<var>n</var>) times in the sequence. The values of G(<var>n</var>) for the first few <var>n</var> are</p>
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<table cellspacing='1' cellpadding='5' border='0' align='center'>
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<tr><td align='left'><var>n</var></td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td><td>14</td><td>15</td><td>…</td></tr>
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<tr><td>G(<var>n</var>)</td><td>1</td><td>2</td><td>2</td><td>3</td><td>3</td><td>4</td><td>4</td><td>4</td><td>5</td><td>5</td><td>5</td><td>6</td><td>6</td><td>6</td><td>6</td><td>…</td></tr></table></div></p>
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<p>You are given that G(10<sup>3</sup>) = 86, G(10<sup>6</sup>) = 6137.<br />
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You are also given that ΣG(<var>n</var><sup>3</sup>) = 153506976 for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 10<sup>3</sup>.</p>
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<p>Find ΣG(<var>n</var><sup>3</sup>) for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 10<sup>6</sup>.</p>
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