Euler's ``exemplum memorabile inductionis fallacis'' and $q$-trinomial coefficients

The trinomial coefficients are defined centrally by $\Sigma _{j = - m}^\infty {(_j^m)_2}{x^j} = {(1 + x + {x^{ - 1}})^m}$. Euler observed that for $- 1 \leq m \leq 7$, $3{(_{\,\;0}^{m + 1})_2} - {(_{\,\;0}^{m + 2})_2} = {F_m}({F_m} + 1)$, where ${F_m}$ is the $m$th Fibonacci number. The assertion is false for $m > 7$. We prove general identities--one of which reduces to Euler's assertion for $m \leq 7$. Our main object is to analyze $q$-analogs extending Euler's observation. Among other things we are led to finite versions of dissections of the Rogers-Ramanujan identities into even and odd parts.