Triangle Tiling

$\begin{figure}\begin{center}\BoxedEPSF{TriangleTilings.epsf scaled 551}\end{center}\end{figure}$

The total number of triangle (including inverted ones) in the above figures is given by

$\begin{displaymath} N(n)=\cases{ {\textstyle{1\over 8}}n(n+2)(2n+1) & for $n$\ ... ...cr {\textstyle{1\over 8}}[n(n+2)(2n+1)-1] & for $n$\ odd.\cr} \end{displaymath}$

The first few values are 1, 5, 13, 27, 48, 78, 118, 170, 235, 315, 411, 525, 658, 812, 988, 1188, 1413, 1665, ... (Sloane's A002717).

References

Conway, J. H. and Guy, R. K. ``How Many Triangles.'' In The Book of Numbers. New York: Springer-Verlag, pp. 83-84, 1996.

Sloane, N. J. A. Sequence A002717/M3827 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.