User:Wguynes/Proofs

Derivation of triangular number test function

Q: What triangular numbers are also tetrahedral numbers?

Let T_n = nth triangular number.

Let T_m = mth tetrahedral number.

T_n = T_m where

$\left\{n={\frac {-3+{\sqrt {12m(m+1)(m+2)+9}}}{6}}\quad |\quad n\in \mathbb {Z} ^{+},m\in \mathbb {Z} ^{+}\right\}$

$\left\{n={\sqrt {{\tfrac {1}{3}}m^{3}+m^{2}+{\tfrac {2}{3}}m+{\tfrac {1}{4}}}}-{\tfrac {1}{2}}\quad |\quad n\in \mathbb {Z} ^{+},m\in \mathbb {Z} ^{+}\right\}$