  The realistic models of neutron stars are considered for simple $R+\alpha
R^2$ gravity and equivalent Brance-Dicke theory with dilaton field in Einsein
frame. For negative values of $\alpha$ we have no acceptable results from
astrophysical viewpoint: the resulting solution for spherical stars doesn't
coincide with Schwarzschild solution on spatial infinity. The mass of star from
viewpoint of distant observer tends to very large values. For $\alpha>0$ it is
possible to obtain solutions with required asymptotics and well-defined star
mass. The mass confined by stellar surface decreases with increasing of
$\alpha$ but we have some contribution to mass from gravitational sphere
appearing outside the star. The resulting effect is increasing of gravitational
mass from viewpoint of distant observer. But another interpretation take place
in a case of equivalent Brance-Dicke theory with massless dilaton field in
Einstein frame. The mass of star increases due to contribution of dilaton field
inside the star. We also considered the possible constraints on $R^{2}$ gravity
from GW 170817 data. According to results of Bauswein et al. the lower limit on
threshold mass is $2.74^{+0.04}_{-0.01}$ $M_{\odot}$. This allows to exclude
some equations of state for dense matter. But in $R^2$ gravity the threshold
mass increases for given equation of state with increasing of $\alpha$. In
principle it can helps in future discriminate between General Relativity and
square gravity (of course one need to know equation of state with more accuracy
rather than now.
