a class of artinian local rings of homogeneous type

‎let $i$ be an ideal in a regular local ring $(r,n)$‎, ‎we will find‎ ‎bounds on the first and the last betti numbers of‎ ‎$(a,m)=(r/i,n/i)$‎. ‎if $a$ is an artinian ring of the embedding‎ ‎codimension $h$‎, ‎$i$ has the initial degree $t$ and $mu(m^t)=1$‎, ‎we call $a$ a {it $t-$extended stretched local ring}‎. ‎this class of‎ ‎local rings is a natural generalization of the class of stretched local rings studied by sally‎, ‎elias and valla‎. ‎for a $t-$extended stretched local ring‎, ‎we show that ${h+t-2choose t-1}-h+1leq tau(a)leq {h+t-2choose‎ ‎t-1}$ and $ {h+t-1choose t}-1 leq mu(i) leq {h+t-1choose t}$‎. ‎moreover $tau(a)$ reaches the upper bound if and only if $mu(i)$‎ ‎is the maximum value‎. ‎using these results‎, ‎we show when‎ ‎$beta_i(a)=beta_i(gr_m(a))$ for each $igeq 0$‎. ‎beside‎, ‎we will‎ ‎investigate the rigid behavior of the betti numbers of $a$ in the‎ ‎case that $i$ has initial degree $t$ and $mu(m^t)=2$‎. ‎this class‎ ‎is a natural generalization of {it almost stretched local rings}‎ ‎again studied by elias and valla‎. ‎our research extends several‎ ‎results of two papers by rossi‎, ‎elias and valla‎.