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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.number_series.tangent_numbers"></a><a class="link" href="tangent_numbers.html" title="Tangent Numbers">Tangent Numbers</a>
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</h3></div></div></div>
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<p>
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<a href="http://en.wikipedia.org/wiki/Tangent_numbers" target="_top">Tangent numbers</a>,
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also called a zag function. See also <a href="http://mathworld.wolfram.com/TangentNumber.html" target="_top">Tangent
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number</a>.
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</p>
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<p>
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The first few values are 1, 2, 16, 272, 7936, 353792, 22368256, 1903757312
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... (sequence <a href="http://oeis.org/A000182" target="_top">A000182 in OEIS</a>).
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They are called tangent numbers because they appear as numerators in the
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Maclaurin series of <code class="computeroutput"><span class="identifier">tan</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span></code>.
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</p>
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<p>
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<span class="bold"><strong>Important:</strong></span> there are two competing definitions
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of Tangent numbers in common use (depending on whether you take the even
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or odd numbered values as non-zero), we use:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/tangent_number_def.svg"></span>
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</p></blockquote></div>
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<p>
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Which gives:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/tangent_number_def2.svg"></span>
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</p></blockquote></div>
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<p>
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Tangent numbers are used in the computation of Bernoulli numbers, but are
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also made available here.
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</p>
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<h5>
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<a name="math_toolkit.number_series.tangent_numbers.h0"></a>
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<span class="phrase"><a name="math_toolkit.number_series.tangent_numbers.synopsis"></a></span><a class="link" href="tangent_numbers.html#math_toolkit.number_series.tangent_numbers.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">detail</span><span class="special">/</span><span class="identifier">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">T</span> <span class="identifier">tangent_t2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">);</span> <span class="comment">// Single tangent number (default policy).</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
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<span class="identifier">T</span> <span class="identifier">tangent_t2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span> <span class="special">&</span><span class="identifier">pol</span><span class="special">);</span> <span class="comment">// Single tangent number (user policy).</span>
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<span class="comment">// Multiple tangent numbers (default policy).</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">></span>
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<span class="identifier">OutputIterator</span> <span class="identifier">tangent_t2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_tangent_t2n</span><span class="special">,</span>
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<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span>
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<span class="comment">// Multiple tangent numbers (user policy).</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">OutputIterator</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
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<span class="identifier">OutputIterator</span> <span class="identifier">tangent_t2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_tangent_t2n</span><span class="special">,</span>
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<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">);</span>
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</pre>
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<h5>
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<a name="math_toolkit.number_series.tangent_numbers.h1"></a>
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<span class="phrase"><a name="math_toolkit.number_series.tangent_numbers.examples"></a></span><a class="link" href="tangent_numbers.html#math_toolkit.number_series.tangent_numbers.examples">Examples</a>
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</h5>
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<p>
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We can compute and save a few Tangent numbers.
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</p>
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<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">float</span><span class="special">></span> <span class="identifier">tn</span><span class="special">;</span> <span class="comment">// Space for some `float` precision Tangent numbers.</span>
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<span class="comment">// Start with Bernoulli number 0.</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tangent_t2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>(</span><span class="number">1</span><span class="special">,</span> <span class="number">6</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">back_inserter</span><span class="special">(</span><span class="identifier">tn</span><span class="special">));</span> <span class="comment">// Fill vector with even Tangent numbers.</span>
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<span class="keyword">for</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="identifier">tn</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span> <span class="identifier">i</span><span class="special">++)</span>
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<span class="special">{</span> <span class="comment">// Show vector of even Tangent numbers, showing all significant decimal digits.</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span>
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<span class="special"><<</span> <span class="string">" "</span>
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<span class="special"><<</span> <span class="identifier">tn</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
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<span class="special">}</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
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</pre>
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<p>
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The output is:
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</p>
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<pre class="programlisting"><span class="number">1</span> <span class="number">2</span> <span class="number">16</span> <span class="number">272</span> <span class="number">7936</span> <span class="number">353792</span>
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</pre>
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<p>
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The source of this example is at <a href="../../../../example/bernoulli_example.cpp" target="_top">bernoulli_example.cpp</a>
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</p>
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<h5>
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<a name="math_toolkit.number_series.tangent_numbers.h2"></a>
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<span class="phrase"><a name="math_toolkit.number_series.tangent_numbers.implementation"></a></span><a class="link" href="tangent_numbers.html#math_toolkit.number_series.tangent_numbers.implementation">Implementation</a>
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</h5>
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<p>
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Tangent numbers are calculated as intermediates in the calculation of the
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<a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli numbers</a>:
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refer to the <a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli
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numbers</a> documentation for details.
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</p>
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</div>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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