<p>Quote from <span style="white-space:nowrap"><a href="https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="Sloane's"><img alt="Sloane's" src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/11px-OEISicon_light.svg.png" decoding="async" width="11" height="15" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/17px-OEISicon_light.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/22px-OEISicon_light.svg.png 2x" data-file-width="409" data-file-height="556"></a> <a href="//oeis.org/A000045" class="extiw" title="oeis:A000045">A000045</a></span>:
</p>
<pre>F(n) = number of compositions of n into odd parts; 
e.g. F(6) counts 1+1+1+1+1+1, 1+1+1+3, 1+1+3+1, 1+3+1+1, 1+5, 3+1+1+1, 3+3, 5+1.
</pre>
<p>Due to the bijection between <a href="https://en.wikipedia.org/wiki/Composition_(number_theory)" class="extiw" title="w:Composition (number theory)">compositions</a><br><a href="//commons.wikimedia.org/wiki/File:Binary_and_compositions_4.svg" class="image"><img alt="Binary and compositions 4.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Binary_and_compositions_4.svg/500px-Binary_and_compositions_4.svg.png" decoding="async" width="500" height="220" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Binary_and_compositions_4.svg/750px-Binary_and_compositions_4.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Binary_and_compositions_4.svg/1000px-Binary_and_compositions_4.svg.png 2x" data-file-width="660" data-file-height="290"></a><br>
this can be expressed in terms of binary numbers:
</p><p><b>Odd composition addends:</b>
</p>
<ul><li><b>1s</b> match the space between binary 0s, the space left of leftmost 0s, the space right of rightmost 0s</li>
<li><b>3s, 5s, 7s, 9s ...</b> match blocks of binary 1s of length 2, 4, 6, 8 ...</li></ul><p>These blocks, corresponding to odd composition addends, are marked in strong red.
</p><p>Columns without even composition addents (i.e. without light red squares) are marked by a black dot.
</p>
Counting these black dots from left side to a green trigon gives a Fibonacci number.
<table><tbody><tr><td style="padding: 15px;"><a href="//commons.wikimedia.org/wiki/File:Watchduck.svg" class="image"><img alt="Watchduck.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Watchduck.svg/40px-Watchduck.svg.png" decoding="async" width="40" height="46" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Watchduck.svg/60px-Watchduck.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Watchduck.svg/80px-Watchduck.svg.png 2x" data-file-width="703" data-file-height="806"></a>
</td>
<td><a href="//commons.wikimedia.org/wiki/User:Watchduck" title="User:Watchduck">Watchduck</a><br><span style="color: gray; font-size: 80%;">You can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck".</span>
</td></tr></tbody></table>
<span class="int-own-work" lang="en">Own work</span>