Do the hexagonal numbers occur in any geometric contexts OTHER than canonical "Vienna sausage" close-packing?

If so, where?

I have discussed one possible example here:


(See the draft paper in progress . . .)

And I am therefore curious to know whether this example might align with other known contexts for the hexagonal numbers.

  • $\begingroup$ Not a geometric result, but you may enjoy the Fermat polygonal number theorem. $\endgroup$ – Jack D'Aurizio Nov 9 '17 at 18:45
  • $\begingroup$ Thank you so much for the reference, Jack. Since my context crops up specifically when considering a certain kind of "line" in relation to a certain kind of "plane", it may be that my context is not only related to the close-packing "geometric context", but also to the "non-geometric" context which you've kindly pointed out. So, thank you very much again for what MAY turn out to be an important "hint" about deeper things going on .. . . $\endgroup$ – David Halitsky Nov 9 '17 at 18:52
  • $\begingroup$ See OEIS sequence A000384 and links there. I'm not sure which you would consider "geometric". $\endgroup$ – Robert Israel Nov 9 '17 at 19:07
  • $\begingroup$ Thank you so much Robert. Although I'm familiar with OEIS, it never occurred to me to look there (I'm a little embarrassed about this, to tell you the truth.) Also, since Schutzenberger many years ago showed a relationship between power series and some of the objects in the class with which I'm dealing (see link below), I was intrigued by the mention of a power series expansion in one of the links for A000384. So - thanks so much again ! en.wikipedia.org/wiki/… $\endgroup$ – David Halitsky Nov 9 '17 at 19:20

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