User:Spundun/Math

TRICOUNT

For any new level added we need to calculate the # of new upright triangles and new inverted triangles.

Upright Triangles

For any level $l$, The # of new upright triangles is

${\frac {l}{2}}(l+1)$

So the cumulative upright triangles are

$\sum _{l=1}^{L}{\frac {1}{2}}(l^{2}+l)$

$={\frac {1}{2}}\sum _{l=1}^{L}l^{2}+{\frac {1}{2}}\sum _{l=1}^{L}l$

$={\frac {1}{2}}({\frac {2L^{3}+3L^{2}+L}{6}}+{\frac {L^{2}+L}{2}})$

$={\frac {1}{2}}({\frac {2L^{3}+3L^{2}+L}{6}}+{\frac {3L^{2}+3L}{6}})$

$={\frac {1}{2}}{\frac {2L^{3}+6L^{2}+4L}{6}}$

$={\frac {L^{3}+3L^{2}+2L}{6}}$

Inverted Triangles

For any level $l$, the biggest inverted triangle will have sides $w_{max}=\lfloor {\frac {l}{2}}\rfloor$

For both $l=2,l=3,w_{max}=1:l=4,l=5,w_{max}=2$ ... and so on.

The # of inverted triangles over the level $l$ of side $w$ will be

So the total # of inverted triangles introduced by a level $l$ will be $\sum _{w=1}^{w_{\mathrm {max} }}l-2w+1$

$=lw_{max}-2({\frac {w_{\max }}{2}}(w_{max}+1))+w_{max}$

$=lw_{\max }-w_{max}(w_{max}+1)+w_{max}$

$=lw_{\max }-w_{\max }^{2}$

$w_{\max }$ is a function of $l$ so we can write it as $w_{\max _{l}}$

So total accumulated inverted triangles over all the levels is

$\sum _{l=1}^{L}lw_{\max _{l}}-w_{\max _{l}}^{2}$

Expressing $l$ in terms of $w_{\max _{l}}$

$l=2w_{\max _{l}}$ if $l$ is even

$l=2w_{\max _{l}}+1$ if $l$ is odd

an even and odd pair would have the same $w_{\max _{l}}$

So the summation can be written as

$\sum _{w=0}^{\lfloor {\frac {L-1}{2}}\rfloor }w(2w)-w^{2}+w(2w+1)-w^{2}$ plus (if $L$ is even) $L({\frac {L}{2}})-({\frac {L}{2}})^{2}$

$=\sum _{w=0}^{\lfloor {\frac {L-1}{2}}\rfloor }2w^{2}+w$ plus (if $L$ is even) ${\frac {L^{2}}{4}}$

$=2\sum _{w=0}^{\lfloor {\frac {L-1}{2}}\rfloor }w^{2}+\sum _{w=0}^{\lfloor {\frac {L-1}{2}}\rfloor }w$ plus (if $L$ is even) ${\frac {L^{2}}{4}}$

$=2{\frac {\lfloor {\frac {L-1}{2}}\rfloor (\lfloor {\frac {L-1}{2}}\rfloor +1)(2\lfloor {\frac {L-1}{2}}\rfloor +1)}{6}}+{\frac {\lfloor {\frac {L-1}{2}}\rfloor (\lfloor {\frac {L-1}{2}}\rfloor +1)}{2}}$ plus (if $L$ is even) ${\frac {L^{2}}{4}}$

$={\frac {(\lfloor {\frac {L-1}{2}}\rfloor ^{2}+\lfloor {\frac {L-1}{2}}\rfloor )(2\lfloor {\frac {L-1}{2}}\rfloor +1)}{3}}+{\frac {\lfloor {\frac {L-1}{2}}\rfloor ^{2}+\lfloor {\frac {L-1}{2}}\rfloor }{2}}$ plus (if $L$ is even) ${\frac {L^{2}}{4}}$

$={\frac {\lfloor {\frac {L-1}{2}}\rfloor ^{2}+\lfloor {\frac {L-1}{2}}\rfloor +2\lfloor {\frac {L-1}{2}}\rfloor ^{3}+2\lfloor {\frac {L-1}{2}}\rfloor ^{2}}{3}}+{\frac {\lfloor {\frac {L-1}{2}}\rfloor ^{2}+\lfloor {\frac {L-1}{2}}\rfloor }{2}}$ plus (if $L$ is even) ${\frac {L^{2}}{4}}$

$={\frac {2\lfloor {\frac {L-1}{2}}\rfloor ^{3}+3\lfloor {\frac {L-1}{2}}\rfloor ^{2}+\lfloor {\frac {L-1}{2}}\rfloor }{3}}+{\frac {\lfloor {\frac {L-1}{2}}\rfloor ^{2}+\lfloor {\frac {L-1}{2}}\rfloor }{2}}$ plus (if $L$ is even) ${\frac {L^{2}}{4}}$

$={\frac {4\lfloor {\frac {L-1}{2}}\rfloor ^{3}+9\lfloor {\frac {L-1}{2}}\rfloor ^{2}+5\lfloor {\frac {L-1}{2}}\rfloor }{6}}$ plus (if $L$ is even) ${\frac {L^{2}}{4}}$