Let $b(n)$ be the largest power of 2 that divides $n$. For example $b(24)=8$.

Define the recursive function: $$\begin{array}{r}\begin{array}{rl}A(0)& =1\\ A(2n)& =3A(n)+5A(2n-b(n))n>0\\ A(2n+1)& =A(n)\end{array}\end{array}$$ and let $H(t,r)=A((2^t+1{)}^r)$.

You are given $H(3,2)=A(81)=636056$.

Find $H({10}^{14}+31,62)$. Give your answer modulo $1000062031$.