Table 1. Results for X chromosomes in the $2^n$-way RIL by sibling mating in the last generation $g=U+V+1$, where $U=0$ for $n=1$ and $U=n-2$ for $n\ge 2$.

Quantity	Theoretical Prediction
(A) 2 ways sibling
${\alpha }_g^{mp}\left(12\right)$	$\frac{5+\sqrt{5}}{10}{\left({\lambda }_1\right)}^V+conjugate$
$R_g^X$	$\frac{8}{3}-\frac{20+8\sqrt{5}}{15}{\left({\lambda }_1\right)}^V+conjugate$
$R_g^m$	$\frac{8}{3}-\frac{2}{3}{\left(-\frac{1}{2}\right)}^V-\frac{5+3\sqrt{5}}{5}{\left({\lambda }_1\right)}^V+conjugate$
$J_g^{mp}\left(1122\right)$	$\frac{8}{3}+\frac{1}{3}{\left(-\frac{1}{2}\right)}^V-\left(\frac{3+\sqrt{5}}{2}+\frac{5+\sqrt{5}}{10}V\right){\left({\lambda }_1\right)}^V+conjugate$
$J_g^{mp+}\left(1232\right)$	$0$
$J_g^{mp-}\left(1232\right)$	0
(B) $2^n\left(n\ge 2\right)$ ways sibling
${\alpha }_g^{mp}\left(12\right)$	$\frac{5+3\sqrt{5}}{10}{\left({\lambda }_1\right)}^V+conjugate$
$R_g^X$	$\frac{2}{3}\left(U+6\right)-\frac{30+14\sqrt{5}}{15}{\left({\lambda }_1\right)}^V+conjugate$
$R_g^m$	$\frac{2}{3}\left(U+6\right)+\frac{1}{3}\left(R_{U+1}^m-R_{U+1}^p\right){\left(-\frac{1}{2}\right)}^V-\frac{10+4\sqrt{5}}{5}{\left({\lambda }_1\right)}^V+conjugate$
$J_g^{mp}\left(1122\right)$	$\begin{array}{l}\frac{2}{3}\left(U+6\right)-\frac{1}{6}\left(R_{U+1}^m-R_{U+1}^p\right){\left(-\frac{1}{2}\right)}^V\\ -\left[\frac{10+4\sqrt{5}}{5}+\frac{1+\sqrt{5}}{4}{R}_{U+1}^m+\frac{5+\sqrt{5}}{20}{R}_{U+1}^p+\frac{5+3\sqrt{5}}{10}V\right]{\left({\lambda }_1\right)}^V+conjugate\end{array}$
$J_g^{mp+}\left(1232\right)$	$-{\left(\frac{1}{2}\right)}^V+\left[\frac{5+3\sqrt{5}}{10}+\frac{1+\sqrt{5}}{4}{R}_{U+1}^m+\frac{5+\sqrt{5}}{20}{R}_{U+1}^p\right]{\left({\lambda }_1\right)}^V+conjugate$
$J_g^{mp-}\left(1232\right)$	${\left(\frac{1}{2}\right)}^{V+1}+\left(R_{U+1}^p-R_{U+1}^m+1\right){\left(-\frac{1}{2}\right)}^{V+1}$

The eigenvalues ${\lambda }_1=(1+\sqrt{5})/4$ and ${\lambda }_2=(1-\sqrt{5})/4$. The map expansions $R_{U+1}^p$ and $R_{U+1}^m$ are given by equations (11a, 11b). The conjugate is given by replacing $\sqrt{5}$ with $-\sqrt{5}$ from the terms involving ${\lambda }_1$. For example, the conjugate term for $R_g^X$ in (A) is given by $-(20-8\surd 5){\left({\lambda }_2\right)}^V/15$. RIL, recombinant inbred line.