Detereministic solution formulae for elementary CA rules

For many cellular automata, you can express the state of a given cell after \(n\) iterations starting from initial configuration \(x\) by an explicit formula. The formulae below come from the "Solvable cellular automata" book (click on the image on the right).

Symbol \([F^n_w(x)]_j\) represents the state of cell \(j\) after \(n\) iterations or rule \(F\) with Wolfram number \(w\). Initial condition is represented by \(x\), so that \(x_i\) is the state of cell \(i\) in the initial configuration. For elementary CA, \(x_i \in \{0,1\}\).

Special notation used in solution formulae: $$\bar{x}=1-x$$ $$ \mathrm{ev}(n)=\frac{1}{2}(-1)^n+\frac{1}{2}, $$ $$ I_k(n)= \begin{cases} 1& n\leq k, \\ 0 &\text{otherwise.} \end{cases} $$

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Rule 1 \begin{equation*} [F^n_1(x)]_j=\begin{cases} \bar{x}_{j-1} \bar{x}_j \bar{x}_{j+1} & \text{$n$ odd,}\\ (1-\bar{x}_{j-2} \bar{x}_{j-1} \bar{x}_j) (1-\bar{x}_{j-1} \bar{x}_j \bar{x}_{j+1}) (1-\bar{x}_j \bar{x}_{j+1} \bar{x}_{j+2}) & \text{$n$ even.} \end{cases} \end{equation*} Rule 2 \begin{equation*} [F_{2}^n(x)]_j = x_{{j+n}}x_{{j+n-2}}x_{{j+n-1}}-x_{{j+n}}x_{{j+n-2}}-x_{{j+n}}x_{{j+n- 1}}+x_{{j+n}} \end{equation*} Rule 3 \begin{equation*} [F_3^n(x)]_j=\begin{cases} 1-\bar{x}_{j-k-1}\bar{x}_{j-k} -\bar{x}_{j-k} \bar{x}_{j-k+1} +\bar{x}_{j-k-1}\bar{x}_{j-k} \bar{x}_{j-k+1} & \text{$n=2k,$}\\ x_{{j-k-1}}x_{{j-k}}-x_{{j-k}}-x_{{j-k-1}}+1 & \text{$n=2k+1.$} \end{cases} \end{equation*} Rule 4 \begin{equation*} [F_{4}^n(x)]_j = x_{{j-1}}x_{{j}}x_{{j+1}}-x_{{j-1}}x_{{j}}-x_{{j}}x_{{j+1}}+x_{{j}} \end{equation*} Rule 5 \begin{equation*} [F^n_5(x)]_j=\begin{cases} 1-x_{{j-1}}-x_{{j+1}}+x_{{j-1}}x_{{j+1}} & \text{$n$ odd,}\\ x_{{j}}+x_{{j-2}}x_{{j+2}}-x_{{j-2}}x_{{j}}x_{{j+2}} & \text{$n$ even.} \end{cases} \end{equation*} Rule 8 \begin{equation*} [F_{8}^n(x)]_i = \begin{cases}-x_{{j-1}}x_{{j}}x_{{j+1}}+x_{{j}}x_{{j+1}}& n=1 \\ 0& \text{otherwise} \end{cases} \end{equation*} Rule 10 \begin{equation*} [F_{10}^n(x)]_j = -x_{{j+n}}x_{{j+n-2}}+x_{{j+n}} \end{equation*} Rule 11 \begin{equation*} [F^n_{11}(x)]_j=[F^{n-1}_{43}(y)]_j \end{equation*} where \begin{equation*} y_j=1+x_{j-1} x_{j}-x_{j-1}-x_{j+1} x_{j-1} x_{j}-x_{j}+x_{j+1} x_{j}. \end{equation*} Rule 12 \begin{equation*} [F_{12}^n(x)]_j = -x_{{j-1}}x_{{j}}+x_{{j}} \end{equation*} Rule 13 \begin{gather*} [F_{13}^n(x)]_j = x_{{j}}+\sum _{r=1}^{n} \left( -1 \right) ^{r}\prod _{i=0}^{r} x_{{j-i}} +\sum _{r=1}^{n} \left( -1 \right) ^{r+1} \prod _{i=-1}^{r} \bar{x}_{{j-i}} \\ +x_{{j+2}} \bar{x}_{{j+1}} \bar{x}_{{j}} \sum _{r=2}^{n} \left( -1 \right) ^{r}\prod _{i=1}^{r}x_{{j-i}} \end{gather*} Rule 14 \begin{equation*} [F^n_{14}(x)]_j=\sum_{\substack{\mathbf{a}\in \{0, 1\}^{2n+1} \\ M_{2n-1}(\mathbf{a})\leq n-1 \\ M_{2n+1}(\mathbf{a})=n }}\Psi_{\mathbf{a}}(x_{j-n}, x_{j-n+1}, \ldots, x_{j+n}), \end{equation*} where \begin{align*} M_0&=0,\\ M_i&=M_{i-1}+\left[ a_i= \frac{1}{2} -\frac{1}{2} (-1)^{\max{\{i-1-M_{i-1},M_{i-1}\}}} \vee i=M_{i-1}+n+2 \right]. \end{align*} Rule 15 \begin{equation*} [F_{15}^n(x)]_j = \frac{1}{2}\, \left( -1 \right) ^{n+1}+\frac{1}{2}+ \left( -1 \right) ^{n}x_{{j-n}} \end{equation*} Rule 19 \begin{equation*} [F_{19}^n(x)]_j = \begin{cases} x_{{j-1}}x_{{j}}x_{{j+1}}-x_{{j-1}}x_{{j+1}}-x_{{j}}+1 & n=1, \\ \frac{1}{2} \, \left( -1 \right) ^{n+1}+\frac{1}{2}+ \left( -1 \right) ^{n}A_j& \text{otherwise,} \end{cases} \end{equation*} \begin{multline*} A_j= -x_{{j}}x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}}+x_{{j}}x_{{j-2}}x_{{j-1}}x_{{j+1}} +x_{{j}}x_{{j-2}}x_{{j-1}}x_{{j+2}}\\+x_{{j}}x_{{j-2}}x_{{j+1}}x_{{j+2}}+x_{{j}}x_{{j-1}}x_{{j+1}}x_{{j+2}}-x_{{j-2}}x_{{j-1}}x_{{j}}-x_{{j}}x_{{j-2}}x_{{j+1}}\\ -x_{{j}}x_{{j-2}}x_{{j+2}} -2\,x_{{j-1}}x_{{j}}x_{{j+1}}-x_{{j}}x_{{j-1}}x_{{j+2}}-x_{{j}}x_{{j+1}}x_{{j+2}}+x_{{j-2}}x_{{j}}\\+x_{{j}}x_{{j-1}}+x_{{j}}x_{{j+1}}+x_{{j}}x_{{j+2}}+x_{{j-1}}x_{{j+1}}. \end{multline*} Rule 23 \begin{multline*} [F_{23}^{n}(x)]_j = x_{{-n+j}}\prod _{i=1}^{n} \bar{x}_{{2\,i-n-1+j}} x_{{2\,i- n+j}}\\+\sum _{k=0}^{n-1} \left( x_{{k+j}} \left( x_{-k-1+j}\bar{x}_{k+1+j} +x_{k+1+j} \right) \prod _{i=0}^{k-1}x_{{-k+2\,i+j}} \bar{x}_{{-k+2\,i+1+j}} \right) \text{\,\, for $n$ even}, \end{multline*} \begin{multline*} [F_{23}^{n}(x)]_j =1-x_{{-n+j}}\prod _{i=1}^{n} \bar{x}_{{2\,i-n-1+j}} x_{{2\,i- n+j}}\\-\sum _{k=0}^{n-1} \left( x_{{k+j}} \left( x_{-k-1+j}\bar{x}_{k+1+j} +x_{k+1+j} \right) \prod _{i=0}^{k-1}x_{{-k+2\,i+j}} \bar{x}_{{-k+2\,i+1+j}} \right) \text{\,\, for $n$ odd}. \end{multline*} Rule 24 \begin{equation*} [F_{24}^n(x)]_j = \begin{cases} -x_{{j-1}}x_{{j}}+x_{{j}}x_{{j+1}}-x_{{j-1}}x_{{j+1}}+x_{{j-1}}& n=1 \\ B_n & \text{otherwise} \end{cases} \end{equation*} \begin{multline*} B_n=x_{{j-n}}-x_{{j-n}}x_{{j-n+1}}x_{{j-n+2}}x_{{j-n+3}}-x_{{j-n}}x_{{j-n+1}}x_{{j-n+2}}x_{{j-n+4}}\\ +x_{{j-n}}x_{{j-n+1}}x_{{j-n+3}}x_{{j-n+4}}+x_{{j-n}}x_{{j-n+2}}x_{{j-n+3}}x_{{j-n+4}}\\-x_{{j-n+1}}x_{{j-n+2}}x_{{j-n+3}}x_{{j-n+4}} +x_{{j-n}}x_{{j-n+1}}x_{{j-n+2}}-x_{{j-n}}x_{{j-n+3}}x_{{j-n+4}}\\+x_{{j-n+1}}x_{{j-n+2}}x_{{j-n+3}}+x_{{j-n+1}}x_{{j-n+2}}x_{{j-n+4}}-x_{{j-n}}x_{{j-n+1}}-x_{{j-n}}x_{{j-n+2}} \end{multline*} Rule 27 \begin{multline*} [F^{n}_{27}(x)]_j=\bar{x}_{{j-k}}+x_{{j-k+1}} \left( x_{{j-k}}-x_{{j-k-1}} \right) \\+\sum _{p =1}^{k} \left( \left( x_{{j-k-2+3\,p}}-x_{{j-k+1+3\,p}} \right) \prod _{i=0}^{p}x_{{j-k-1+3\,i}} \bar{x}_{{j-k+3\,i}} \right), \,\,n=2k+1, \end{multline*} \begin{multline*} [F^{n}_{27}(x)]_j=x_{{j-k}}-x_{{j-k+1}} \bar{x}_{{j-k+2}} \left( x_{{j+3-k}} x_{{j-k-1}}+x_{{j-k}}-x_{{j+3-k}}-x_{{j-k-1}} \right)\\ -\sum _{p=1}^{k- 1} \left( \left( x_{{j-k+3\,p}}-x_{{j+3-k+3\,p}} \right) \bar{x}_{{j-k-1}} \prod _{i=0}^{p}x_{{j-k+1+3\,i}} \bar{x}_{{j-k+2+3\,i}} \right), \,\,n=2k. \end{multline*} Rule 28 \begin{equation*} [F^n_{28}(x)]_j=[F^{n-1}_{156}(y)]_j, \end{equation*} where \begin{equation*} y_j= -2 x_{ j-1} x_{ j}+x_{ j-1}+x_{ j}-x_{ j-1} x_{ j+1}+x_{ j-1} x_{ j} x_{ j+1}. \end{equation*} Rule 29 \begin{equation*} [F^n_{29}(x)]_j=\begin{cases} 1-x_{{j-1}}x_{{j}}+x_{{j}}x_{{j+1}}-x_{{j+1}} & \text{$n$ odd,}\\ x_{{j}}+ \left( x_{{j+1}}-1 \right) x_{{j-1}} \left( x_{{j-2}}x_{{j}}-x_{{j }}x_{{j+2}}-x_{{j-2}}+x_{{j}} \right) & \text{$n$ even.} \end{cases} \end{equation*} Rule 32 \begin{equation*} [F_{32}^n(x)]_j = x_{{j-n}}\prod _{i=1}^{n} \bar{x}_{{2\,i-n-1+j}} x_{{2\,i-n +j}} \end{equation*} Rule 34 \begin{equation*} [F_{34}^n(x)]_j = -x_{{j+n}}x_{{j+n-1}}+x_{{j+n}} \end{equation*} Rule 36 \begin{equation*} [F_{36}^n(x)]_j = \begin{cases} -x_{{j-1}}x_{{j}}-x_{{j}}x_{{j+1}}+x_{{j-1}}x_{{j+1}}+x_{{j}}& n=1, \\ C_n &\text{otherwise,} \end{cases} \end{equation*} \begin{multline*} C_n=x_{{j}}-x_{{j}}x_{{j-2}}x_{{j-1}}x_{{j+1}}-x_{{j}}x_{{j-2}}x_{{j-1}}x_{{j+2}}-x_{{j}}x_{{j-2}}x_{{j+1}}x_{{j+2}}-x_{{j}}x_{{j-1}}x_{{j+1}}x_{{j+2}}\\+x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}}+x_{{j}}x_{{j-2}}x_{{j-1}}+x_{{j}}x_{{j-2}}x_{{j+1}}+x_{{j}}x_{{j-2}}x_{{j+2}}+x_{{j}}x_{{j-1}}x_{{j+1}}\\+x_{{j}}x_{{j-1}}x_{{j+2}}+x_{{j}}x_{{j+1}}x_{{j+2}}-x_{{j-2}}x_{{j}}-x_{{j-1}}x_{{j}}-x_{{j}}x_{{j+1}}-x_{{j}}x_{{j+2}} \end{multline*} Rule 38 \begin{equation*} [F_{38}^n(x)]_j= \begin{cases} x_{j}x_{j-1}x_{j+1}-x_{j}x_{j-1}-2\,x_{j}x_{j+1}+x_{j} +x_{j+1} & \text{$n=1$,}\\ A_n & \text{$n$ even,}\\ B_n & \text{otherwise,} \end{cases} \end{equation*} \begin{multline*} A_n=-x_{j+n}x_{j+n-4}x_{j+n-3}x_{j+n-2}x_{j+n-1}+x_{j+n}x_{j+ n-4}x_{j+n-3}x_{j+n-1}\\+x_{j+n}x_{j+n-3}x_{j+n-2}-x_{j+n}x _{j+n-3}x_{j+n-1}-x_{j+n}x_{j+n-2}+x_{j+n}, \end{multline*} \begin{multline*} B_n=x_{j+n}x_{j+n-4}x_{j+n-3}x_{j+n-2}x_{j+n-1}-x_{j+n}x_{j+n -4}x_{j+n-3}x_{j+n-1}\\-2\,x_{j+n}x_{j+n-3}x_{j+n-2}x_{j+n-1 }-x_{j+n-4}x_{j+n-3}x_{j+n-2}x_{j+n-1}+x_{j+n}x_{j+n-3}x_ {j+n-2}\\+x_{j+n}x_{j+n-3}x_{j+n-1}+2\,x_{j+n}x_{j+n-2}x_{j +n-1}+x_{j+n-4}x_{j+n-3}x_{j+n-1}\\+x_{j+n-3}x_{j+n-2}x_{j+n -1}-x_{j+n}x_{j+n-2}-2\,x_{j+n}x_{j+n-1}-x_{j+n-3}x_{j+n-1 }\\-x_{j+n-2}x_{j+n-1}+x_{j+n}+x_{j+n-1} \end{multline*} Rule 40 \begin{equation*} [F^n_{40}(x)]_j=[F^{n-1}_{168}(y)]_j \end{equation*} where \begin{equation*} y_j=-2 x_{j+1} x_{j-1} x_{j}+x_{j+1} x_{j}+x_{j+1} x_{j-1} \end{equation*} Rule 42 \begin{equation*} [F_{42}^n(x)]_j = -x_{{j+n}}x_{{j+n-2}}x_{{j+n-1}}+x_{{j+n}} \end{equation*} Rule 43 \begin{equation*} [F^n_{43}(x)]_j=\sum_{\substack{\mathbf{a}\in \{0, 1\}^{2n+1} \\ M_{2n+1}(\mathbf{a})=n}}\Psi_{\mathbf{a}}(x_{j-n}, x_{j-n+1}, \ldots, x_{j+n}), \end{equation*} where \begin{align*} M_0&=0,\\ M_i&=M_{i-1}+\left[ a_i= \frac{1}{2} -\frac{1}{2} (-1)^{\min{\{i-1-M_{i-1},M_{i-1}\}}} \vee i=M_{i-1}+n+2 \right]. \end{align*} Rule 44 \begin{multline*} [F^n_{44}(x)]_j= \bar{x}_{{j-2}} \bar{x}_{{j-1}} x_{{j}} +\sum _ {r=1}^{n-1} \left( C_{ r-1} \left(x_{{j-r-2}} \right) \prod _{k=1}^{2+2 \,r}C_{k+r-1} \left( \bar{x}_{{j+k-r-2}} \right) \right) \\ +\prod _{i=-n}^{n} C_{i-n+1} \left( \bar{x}_{{i+j}} \right) +\prod _{i=-n}^{n}C_{i-n+1} \left( \bar{x}_{{i+j-1}} \right), \end{multline*} where $$ C_k(x)=\begin{cases} x & \text{if $k$ divisible by 3},\\ 1-x & \text{otherwise.} \end{cases} $$ Rule 46 \begin{equation*} [F_{46}^n(x)]_j = \begin{cases}-x_{{j-1}}x_{{j}}-x_{{j}}x_{{j+1}}+x_{{j}}+x_{{j+1}}& n=1,\\ D_n&\text{otherwise,} \end{cases} \end{equation*} \begin{multline*} D_n=-x_{{j+n}}x_{{j+n-3}}x_{{j+n-2}}x_{{j+n-1}}-x_{{j+n-4}}x_{{j+n-3}}x_{{j+n-2}}x_{{j+n-1}}\\+x_{{j+n}}x_{{j+n-3}}x_{{j+n-2}}+x_{{j+n}}x_{{j+n-2}}x_{{j+n-1}}+x_{{j+n-4}}x_{{j+n-3}}x_{{j+n-1}}\\+x_{{j+n-3}}x_{{j+n-2}}x_{{j+n-1}}-x_{{j+n}}x_{{j+n-2}}-x_{{j+n}}x_{{j+n-1}}-x_{{j+n-3}}x_{{j+n-1}}\\-x_{{j+n-2}}x_{{j+n-1}}+x_{{j+n}}+x_{{j+n-1}} \end{multline*} Rule 50 \begin{equation*} [F_{50}^n(x)]_j = \frac{1}{2}+ \frac{1}{2} \left( -1 \right) ^{n} +\sum _{i=1}^{n-1} \left( \left( -1 \right) ^{i+n} \prod _{p=-i}^{i}x_{{p+j}} \right) +\sum _{i=0}^{n} \left( \left( -1 \right) ^{i+n+1} \prod _{p=-i}^{i} \bar{x}_{p+j} \right) \end{equation*} Rule 51 \begin{equation*} [F_{51}^n(x)]_j = \frac{1}{2}+ \frac{1}{2} \left( -1 \right) ^{n+1}+ \left( -1 \right) ^{n}x_{{j}} \end{equation*} Rule 56 \begin{equation*} [F^n_{56}(x)]_j=[F^{n-1}_{184}(y)]_j \end{equation*} where \begin{equation*} y_j=x_{j+1} x_{j}+x_{j-1}-x_{j+1} x_{j-1} x_{j}-x_{j-1} x_{j}. \end{equation*} Rule 60 \begin{equation*} [F_{60}^n(x)]_j = \sum _{i=0}^{n}{n\choose i}x_{{i-n+j}} \mod 2 \end{equation*} Rule 72 \begin{equation*} [F_{72}^n(x)]_j = \begin{cases}-2\,x_{{j-1}}x_{{j}}x_{{j+1}}+x_{{j-1}}x_{{j}}+x_{{j}}x_{{j+1}}& n=1\\ E_n&\text{otherwise} \end{cases} \end{equation*} \begin{multline*} E_n=x_{{j}}x_{{j-2}}x_{{j-1}}x_{{j+1}}+x_{{j}}x_{{j-1}}x_{{j+1}}x_{{j+2}}-x_{{j-2}}x_{{j-1}}x_{{j}}-2\,x_{{j-1}}x_{{j}}x_{{j+1}}\\-x_{{j}}x_{{j+1}}x_{{j+2}}+x_{{j-1}}x_{{j}}+x_{{j}}x_{{j+1}} \end{multline*} Rule 76 \begin{equation*} [F_{76}^n(x)]_j = -x_{{j-1}}x_{{j}}x_{{j+1}}+x_{{j}} \end{equation*} Rule 77 \begin{equation*} [F_{77}^n(x)]_j = x_{{j}}+\sum _{r=1}^{n} \left( -1 \right) ^{r}\prod _{i=-r}^{r }x_{{j+i}} +\sum _{r=1}^{n} \left( -1 \right) ^{r+1} \prod _{i=-r}^{r}\bar{x}_{{j+i}} \end{equation*} Rule 78 \begin{equation*} [F^n_{78}(x)]_j=1- {x}_{j-1} \bar{x}_{j}-\prod _{i=n-1}^{2\,n}\bar{x}_ {{i-n+j}} -S^{(A)}_{n,j} - S^{(B)}_{n,j}+C_{n,j}, \end{equation*} where \begin{multline*} S^{(A)}_{n,j}=\sum_{k=2}^{n-1} \left( \mathrm{ev}(k) x_{j+k-1} \bar{x}_{j+k} \prod_{j-2}^{j-2+k} \bar{x}_i \right) \\ +\sum_{k=1}^{n-1} \left( \mathrm{ev}(k+1) x_{j+k-1} x_{j+k} \bar{x}_{j+k+1} \prod_{j-2}^{j-2+k} \bar{x}_i \right) \\ +\sum_{k=2}^{n-1} \left( \mathrm{ev}(k) x_{j+k-1} x_{j+k} x_{j+k+1} \prod_{j-2}^{j-2+k} \bar{x}_i \right), \end{multline*} \begin{equation*} S^{(B)}_{n,j}=x_{j-1}x_{j}x_{j+1}, \end{equation*} \begin{multline*} C_{n,j}= x_{j-3} x_{j-2} x_{j-1} x_{j} x_{j+1} \sum_{r=2}^{2\lfloor n/2 \rfloor -2 } \left( (-1)^r \prod_{k=2}^rx_{j+k} \right) -\prod_{i=-2}^n x_{j+i} \\ +\bar{x}_{j-3}x_{j-2} x_{j-1} x_{j} x_{j+1} + I_3(n) x_{j-3} x_{j-2} x_{j-1} x_{j} x_{j+1}. \end{multline*} Rule 90 \begin{equation*} [F_{90}^n(x)]_j = \sum _{i=0}^{n}{n\choose i}x_{{2\,i-n+j}} \mod 2 \end{equation*} Rule 105 \begin{equation*} [F_{105}^n(x)]_j=\begin{cases} \displaystyle \sum _{i=0}^{2\,n} {n\choose i-n}_2 x_{{i+j-n}} \mod 2 & \text{$n$ even,}\\ \displaystyle 1 -\sum _{i=0}^{2\,n} {n\choose i-n}_2 x_{{i+j-n}} \mod 2 & \text{$n$ odd.} \end{cases} \end{equation*} Rule 108 \begin{equation*} [F_{108}^n(x)]_j = \begin{cases}-2\,x_{{j}}x_{{j-1}}x_{{j+1}}+x_{{j-1}}x_{{j+1}}+x_{{j}}& n=1,\\ A & \text{$n$ even,}\\ B & \text{otherwise,} \end{cases} \end{equation*} \begin{multline*} A=x_{{j}}-x_{{j}} \left( x_{{j-2}}x_{{j+1}}+x_{{j-2}}x_{{j+2}}+x_{{j-1}}x_{{j+2}} \right)\\ -2\,x_{{j}}x_{{j-2}}x_{{j+2}} \left( x_{{j-1}}x_{{j+1}}-x_{{j-1}}-x_{{j+1}} \right) \end{multline*} \begin{multline*} B=-4\,x_{{j}}x_{{j-3}}x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}}x_{{j+3}}+2\,x _{{j}}x_{{j-3}}x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}}\\ +2\,x_{{j}}x_{{j-3} }x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+3}} +2\,x_{{j}}x_{{j-3}}x_{{j-1}}x_{{ j+1}}x_{{j+2}}x_{{j+3}}\\ +2\,x_{{j}}x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}} x_{{j+3}} +4\,x_{{j-3}}x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}}x_{{j+3}}\\ -2\,x_{{j}}x_{{j-3}}x_{{j-2}}x_{{j-1}}x_{{j+1}}-x_{{j}}x_{{j-3}}x_{{j-1} }x_{{j+1}}x_{{j+2}}-x_{{j}}x_{{j-3}}x_{{j-1}}x_{{j+1}}x_{{j+3}}\\ -4\,x_{{j}}x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}}-x_{{j}}x_{{j-2}}x_{{j-1}}x_{{ j+1}}x_{{j+3}}-2\,x_{{j}}x_{{j-1}}x_{{j+1}}x_{{j+2}}x_{{j+3}}\\ -2\,x_{{j -3}}x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}}-2\,x_{{j-3}}x_{{j-2}}x_{{j-1} }x_{{j+1}}x_{{j+3}}\\ -2\,x_{{j-3}}x_{{j-1}}x_{{j+1}}x_{{j+2}}x_{{j+3}}-2 \,x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}}x_{{j+3}}+x_{{j}}x_{{j-3}}x_{{j- 1}}x_{{j+1}}\\ +2\,x_{{j}}x_{{j-2}}x_{{j-1}}x_{{j+1}}+2\,x_{{j}}x_{{j-2}} x_{{j-1}}x_{{j+2}}+2\,x_{{j}}x_{{j-2}}x_{{j+1}}x_{{j+2}}\\ +2\,x_{{j}}x_{{j-1}}x_{{j+1}}x_{{j+2}}+x_{{j}}x_{{j-1}}x_{{j+1}}x_{{j+3}}+2\,x_{{j-3 }}x_{{j-2}}x_{{j-1}}x_{{j+1}}\\ +x_{{j-3}}x_{{j-1}}x_{{j+1}}x_{{j+2}}+x_{ {j-3}}x_{{j-1}}x_{{j+1}}x_{{j+3}}+x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+2}} +x_{{j-2}}x_{{j-1}}x_{{j+1}}x_{{j+3}}\\ +2\,x_{{j-1}}x_{{j+1}}x_{{j+2}}x_ {{j+3}}-x_{{j}}x_{{j-2}}x_{{j+1}}-x_{{j}}x_{{j-2}}x_{{j+2}}-2\,x_{{j}} x_{{j-1}}x_{{j+1}}-x_{{j}}x_{{j-1}}x_{{j+2}}\\-x_{{j-3}}x_{{j-1}}x_{{j+1 }}-x_{{j-2}}x_{{j-1}}x_{{j+1}}-x_{{j-1}}x_{{j+1}}x_{{j+2}}-x_{{j-1}}x_ {{j+1}}x_{{j+3}}+x_{{j-1}}x_{{j+1}}+x_{{j}} \end{multline*} Rule 128 \begin{equation*} [F_{128}^n(x)]_j = \prod _{i=-n}^{n}x_{{i+j}} \end{equation*} Rule 130 \begin{multline*} [F_{130}^n(x)]_j = \prod _{i=1}^{2\,n+1}x_{{i+j-n-1}} \\ +\sum _{i=0}^{n-1} \left( \left( \frac{1}{2}+\frac{1}{2} \left( -1 \right) ^{i}- \left( -1 \right) ^{i}x_{{n-2-2\,i+j }} \right) \bar{x}_{{n-2\,i+j-1}} x_{{n-2\,i+j}}\prod _{p= 2\,n+2-2\,i}^{2\,n+1}x_{{p+j-n-1}} \right) \end{multline*} Rule 132 \begin{equation*} [F_{132}^n(x)]_j = x_{{j}}-\sum _{r=1}^{n} \left( \left( x_{{j-r}}-x_{{j+r}} \right) ^{2 }\prod _{i=-r+1}^{r-1}x_{{i+j}} \right) \end{equation*} Rule 136 \begin{equation*} [F_{136}^n(x)]_j = \prod _{i=0}^{n}x_{{i+j}} \end{equation*} Rule 138 \begin{equation*} [F_{138}^n(x)]_j = x_{{j+n}}x_{{j+n-2}}x_{{j+n-1}}-x_{{j+n}}x_{{j+n-2}}+x_{{j+n}} \end{equation*} Rule 140 \begin{equation*} [F_{140}^n(x)]_j = \left( 1-x_{{j-1}} \right) x_{{j}}+\prod _{i=n-1}^{2\,n}x_{{i-n+j}} \end{equation*} Rule 142 \begin{equation*} [F^n_{142}(x)]_j=\sum_{\substack{\mathbf{a}\in \{0, 1\}^{2n+1} \\ M_{2n+1}(\mathbf{a})=n}}\Psi_{\mathbf{a}}(x_{j-n}, x_{j-n+1}, \ldots, x_{j+n}), \end{equation*} where \begin{align*} M_0&=0,\\ M_i&=M_{i-1}+\left[ a_i= \frac{1}{2} -\frac{1}{2} (-1)^{\max{\{i-1-M_{i-1},M_{i-1}\}}} \vee i=M_{i-1}+n+2 \right]. \end{align*} Rule 150 \begin{equation*} [F_{150}^n(x)]_j = \sum _{i=0}^{2\,n}T \left( n,i \right) x_{{i+j-n}} \mod 2 \end{equation*} \begin{equation*} T(n,i)=\sum _{p=0}^{n}{n\choose p}{p\choose i-p} \end{equation*} Rule 156 \begin{equation*} [F^n_{156}(x)]_j= \bar{x}_{{j-1}} x_{{j}}+\prod _{i=n-1}^{2\,n}x_ {{i-n+j}} + D_{n,j} + B^{(0)}_{n,j} + B^{(0)}_{n,j} + S^{(0)}_{n,j} + S^{(1)}_{n,j}, \end{equation*} where \begin{equation*} D_{n,j}=x_{{j-n}}\prod _{i=n-1}^{2\,n-1}\bar{x}_{{-i+n+j}}. \end{equation*} \begin{multline*} B^{(0)}_{n,j}=\mathrm{ev} \left( n \right) \sum _{r=2}^{n} \left( \mathrm{sel } \left( r+1 \right) x_{{j-r}}x_{{j-r+1}} x_{{2+j}} \prod _{m=j-r+2}^{1+j}\bar{x}_{{m }} \right) \\+\mathrm{ev} \left( n+1 \right) \sum _{r=2}^{n} \left( \mathrm{ev} \left( r \right) \bar{x}_{{j-r}} x_{{j-r +1}} x_{{2+j}} \prod _{m=j-r+2}^{1+j}\bar{x}_{{m}} \right), \end{multline*} \begin{multline*} B^{(1)}_{n,j}=\mathrm{ev} \left( n+1 \right) \sum _{m=0}^{n-1} \left( \mathrm{ev} \left( m+1 \right) \bar{x}_{{j-2}} \bar{x}_{{j+m+1}} \bar{x}_{{j+m+2}} \prod _{k=-1}^{m}x_{{j+k}} \right)\\ +\mathrm{ev} \left( n \right) \sum _{m=0}^{n-1} \left( \mathrm{ev} \left( m \right) \bar{x}_{{j-2}} \bar{x}_{{j+m+1}} x_{{j+m+2}} \prod _{k=-1}^{m}x_{{j+k}} \right), \end{multline*} \begin{multline*} S^{(0)}_{n,j}=\sum _{k=1}^{n} \left( \mathrm{ev} \left( k \right) x_{{j-k} }x_{{j-k+1}}\prod _{m=j-k+2}^{1+j}\bar{x}_{{m}} \right)\\ +\sum _{k=2}^{n} \left( \mathrm{ev} \left( k+1 \right) \bar{x}_{{j-k}} x_{{j -k+1}}\prod _{m=j-k+2}^{1+j}\bar{x}_{{m}} \right), \end{multline*} \begin{multline*} S^{(1)}_{n,j}=\sum _{k=2}^{n} \left( \left( -1 \right) ^{k} \left( x_{{j+k}}+\mathrm{ev} \left( k+1 \right) \left( x_{{j-1+k}}-1 \right) \right) \prod _{i=j-2}^{j-2+k}x_{{i}} \right)\\ -\sum _{k=2}^{n} \left( x_{{j}}x_{{j-2}}x_{{j-1}} \left( -1 \right) ^{k}\prod _{m=1}^{ k}x_{{j+m}} \right). \end{multline*} Rule 160 \begin{equation*} [F_{160}^n(x)]_j = \prod _{i=0}^{n}x_{{2\,i+j-n}} \end{equation*} Rule 162 \begin{equation*} [F_{162}^n(x)]_j = x_{{j+n}}+\sum _{p=-n+1}^{n} \left( \left( -1 \right) ^{p+n}\prod _{i =j-p}^{j+n}x_{{i}} \right) \end{equation*} Rule 164 \begin{multline*} [F_{164}^n]_j= \prod _{p=0}^{n}x_{j-n+2p} -\prod _{p=-n}^{n}x_{j+p} + \left( 1+\sum _{p=0}^{n} {n\choose p} x_{j-n+2p} \text{\,\,mod 2} \right) \prod _{p=0}^{n-1}x_{j-n+2p+1}\\ +\sum _{i=1}^{n-1} \Bigg[ \left( 1- \prod _{p=0}^{i}x_{j-i+2p} \right) \bar{x}_{j-1-i} \bar{x}_{j+i+1} \\ \times \left( \prod _{p=0}^{i-1}x_{j-i+2p+1} \right) \left( 1+\sum _{p=0}^{i} {i \choose p} x_{j-i+2p} \text{\,\,mod 2} \right) \Bigg] \end{multline*} Rule 168 \begin{multline*} [F_{168}^n(x)]_j \\ = \sum _{k=n+2}^{2\,n} \Biggl( H_k \, \bar{x}_{{n-k+j}} \bar{x}_{{n-k+1+j}} x_{{n-k+2+j}}x_{{n+j}} \prod _{i=2\,n-k+4}^{2\,n} (1- \bar{x}_{{i-n-1+j}} \bar{x}_{{i-n+j}} ) \Biggr) \\ +x_{{n+j}}\prod _{i=1}^{2\,n-1}(1- \bar{x}_{{i-n-1+j}} \bar{x}_{{i-n+j}} ) \end{multline*} \begin{equation*} H_{k}= \begin{cases} 1& \displaystyle \text{if\,\,} \sum _{i=2\,n-k+4}^{2\,n}x_{{i-n-1+j}} \geq n-1, \\ 0 & \text{otherwise.} \end{cases} \end{equation*} Rule 170 \begin{equation*} [F_{170}^n(x)]_j = x_{{j+n}} \end{equation*} Rule 172 \begin{multline*} [F_{172}^n(x)]_j = \bar{x}_{{j-2}} \bar{x}_{{j-1}} x_{{j}}+ \left( \bar{x}_{{j+n-2}} x_{{j+n-1}}+x_{{j+n-2}}x_{{j+n}} \right) \prod _{i=j-2}^{j+n-3}(1- \bar{x}_{{i}} \bar{x} _{{i+1}} ) \end{multline*} Rule 178 \begin{multline*} [F_{178}^n(x)]_j=\begin{cases} \displaystyle x_{{j}}+\sum _{r=1}^{n} \left( -1 \right) ^{r}\prod _{i=-r}^{r }x_{{j+i}} +\sum _{r=1}^{n} \left( -1 \right) ^{r+1} \prod _{i=-r}^{r}\bar{x}_{{j+i}} & \text{$n$ even,}\\ \displaystyle 1 -x_{{j}}-\sum _{r=1}^{n} \left( -1 \right) ^{r}\prod _{i=-r}^{r }x_{{j+i}} -\sum _{r=1}^{n} \left( -1 \right) ^{r+1} \prod _{i=-r}^{r}\bar{x}_{{j+i}} & \text{$n$ odd.} \end{cases} \end{multline*} Rule 184 \begin{equation*} [F^n_{184}(x)]_j=\sum_{\substack{\mathbf{a}\in \{0, 1\}^{2n+1} \\ M_{2n+1}(\mathbf{a})=n}}\Psi_{\mathbf{a}}(x_{j-n}, x_{j-n+1}, \ldots, x_{j+n}), \end{equation*} where \( \Psi \) is the density polynomial and \begin{equation*} M_{i}(\mathbf{a})= \begin{cases} 0 & \text{if $i=0$}, \\ M_{i-1}(\mathbf{a})+1 & \text{if $a_{i}=0$ and $ i \leq 2M_{i-1}+1$}, \\ M_{i-1}(\mathbf{a})+1 & \text{if $a_{i}=1$ and $ i > 2M_{i-1}+1$}, \\ M_{i-1}(\mathbf{a})+1 & \text{if $i=M_{i-1}+n+2$}, \\ M_{i-1}(\mathbf{a}) & \text{otherwise}. \end{cases} \end{equation*} Rule 200 \begin{equation*} [F_{200}^n(x)]_j = -x_{{j-1}}x_{{j}}x_{{j+1}}+x_{{j-1}}x_{{j}}+x_{{j}}x_{{j+1}} \end{equation*} Rule 204 \begin{equation*} [F_{204}^n(x)]_j = x_{{j}} \end{equation*} Rule 232 \begin{multline*} [F_{232}^n(x)]_j = x_{{j-n}}\prod _{i=1}^{n} \bar{x}_{{j+2\,i-n-1}} x_{{j+2\,i -n}} \\ +\sum _{k=1}^{n} x_{{j-k}}x_{{j-k+1}} \left( \prod _{i=1}^{k-1} \bar{x}_{{j-k+2\,i}} x_{{j-k+2\,i+1}} \right) \bar{x}_{{j+k}} \\ +\sum _{k=1}^{n} \left( \prod _{i=0}^{k-2}x_{{j-k+2 \,i+1}} \bar{x}_{{j-k+2\,i+2}} \right) x_{{j+k-1}}x_{{j+k}} \end{multline*}