Combinatorics Problem

by luimichael, Oct 21, 2007, 2:22 AM

http://www.artofproblemsolving.com/Forum/album.php?mode=attach&id=11033
No. of equilateral triangles of the tower with n layers is given by
$ \frac {n(n^2 + 6n - 1)}6$

And what is NEXT?

Maybe generalizing it to 3 dimensional.
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The formula is incorrect as there is a mistake in creating the recursion formula.
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Correction
Let $ S_n$ be the total number of triangles in the tower.
$ S_1 = 1$ ;
$ S_2 = S_1 + 1 + C_2^3$ ;
$ S_3 = S_2 + 2 + C_2^4$;
$ S_4 = S_3 + 3 + 1 + C_2^5$;
$ S_5 = S_4 + 4 + 2 + C_2^6$;
$ S_6 = S_5 + 5 + 3 + 1 + C_2^7$;
$ S_7 = S_6 + 6 + 4 + 2 + C_2^8$;
$ S_8 = S_7 + 7 + 5 + 3 + 1 + C_2^9$.
.............
$ S_{2n} = \frac {n(n + 1)(4n + 1)}2$
and $ S_{2n - 1} = \frac {n(4n^2 - n - 1)}2$.

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