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{\bf Appendix to Cutting 4 by n grids into two congruent pieces}
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{\it Robert Dougherty-Bliss, Natalya Ter-Saakov, and  Doron Zeilberger}

This theorem also follows from the grammar, but was omitted from the paper.

{\bf Theorem 2}: Let $f_a(x)$ be the coefficient of $z^a$ in the Maclaurin expansion (with respect to the variable $z$) of the rational function
$$
\frac{-x^{3}-x^{2}+x +1+\left(x^{5}+2 x^{3}-x^{2}-x \right) z +\left(x^{4}-x^{3}+2 x^{2}-1\right) z^{2}}{z^{3}+z^{2}-3 z +1}
$$
then the number of $4 \times n$ good matrices whose first row starts with $a$  consecutive zeroes (and hence ends with  $n-a$ consecutive ones) is the coefficient of $x^n$ in
the Maclaurin expansion of the rational function
$$
\frac{x^{2 a} f_{a}\! \left(x \right)}{\left(x -1\right) \left(x^{4}+3 x^{2}-1\right)} \quad .
$$



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Robert Dougherty-Bliss, Department of Mathematics, Dartmouth College, 29 N. Main Street, 6188 Kemeny Hall, Hanover NH 03755-3551 \quad .
Email: {\tt robert dot w dot bliss at gmail dot com}


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Natalya Ter-Saakov, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt  nt399 at rutgers dot edu }   \quad .

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Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt DoronZeil at gmail  dot com}   \quad .

{\bf Oct. 15, 2025}

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