Suppose we have a Linear Diophantine Equation $$a_k x_1 + a_{k+1} x_2 + ...+a_{k+n-1}x_n $$ where, $a_k>0 , n>1$ and $x_i>0$
And it is given that $$gcd(a_k, a_{k+1}...a_{k+n-1}) = 1$$
the question i am trying to answer is the number of non-negative unreachable integer solutions for this equation under given conditions.
for example, $$ n=2, a_k = 3, a_{k+1}=4$$ here, $gcd(3, 4)=1$
the equation becomes $$3x_1 + 4x_2$$ possible unreachable integer values, $$1, 2 , 5$$ Is there some kind of formula to answer it for all $n>1$?
Note that there are always finite number of unreachable non-negative integers for $n>1$.