User:Roman Czyborra

Syracuse Bits

$$C^0=LE(1C^0_1C^0_2C^0_3C^0_4\ldots)$$ $$D^1=LE(1C^0_1C^0_2C^0_3C^0_4\ldots)+LE(11C^0_1C^0_2C^0_3\ldots)$$ $$D^1_0=(1+1)\%2=0$$ $$D^1_1=(1+1+C^0_1)\%2=C^0_1$$ $$D^1_2=(1+C^0_1+C^0_2)\%2=(C^0_1==C^0_2)$$ $$D^1_3=(\left\lfloor\frac{1+C^0_1+C^0_2}2\right\rfloor+C^0_2+C^0_3)\%2=((C^0_1\lor C^0_2)\ne C^0_2\ne C^0_3)$$ $$D^1_4=(\left\lfloor\frac{(C^0_1||C^0_2)+C^0_2+C^0_3}2\right\rfloor+C^0_3+C^0_4)\%2 =(C^0_2\lor\left\lfloor\frac{C^0_1+C^0_2+C^0_3}2\right\rfloor+C^0_3+C^0_4)\%2$$ $$D^1_5=(\left\lfloor\frac{C^0_2\lor\left\lfloor\frac{C^0_1+C^0_2+C^0_3}2\right\rfloor+C^0_3+C^0_4}2\right\rfloor+C^0_4+C^0_5)\%2$$

Collatz Recursion

$$c_1=\frac{3c_0+1}{d_1}$$ $$c_2=\frac{3c_1+1}{d_2} =\frac{3\frac{3c_0+1}{d_1}+1}{d_2} =\frac{3^2c_0+3^1+d_1}{d_1d_2}$$ $$c_3=\frac{3c_2+1}{d_3} =\frac{3\frac{3^2c_0+3^1+d_1}{d_1d_2}+1}{d_3} =\frac{3^3c_0+3^2+3^1d_1+d_1d_2}{d_1d_2d_3}$$ $$c_n=\frac{3^nc_0+\sum_{i=0}^{n-1}3^{n-1-i}\prod_{j=1}^id_j}{\prod_{i=1}^n d_i}$$

Collatz Confluence

$$\begin{eqnarray} \\\because\forall^{n\in\mathbb N+1}_{\exists^{[k_0,k_1,k_2,k_3]\in\mathbb N^4}}\begin{bmatrix}2n-0\mapsto1n-0 \\2n-1\mapsto3n-1\end{bmatrix}^{k_0+k_1+2k_2+2k_3+3}_n&=&((2^3+1)^{1}-1)=1 \\\because\forall^{n\in\mathbb N+1}_{\exists^{[k_0,k_1,k_2,k_3]\in\mathbb N^4}}\begin{bmatrix}2n-0\mapsto1n-0 \\2n-1\mapsto3n-1\end{bmatrix}^{k_0+k_1+k_2+1}_n&=&(3^{2+k_2+k_3}-1)=(2+1)^{2+k_2+k_3}-1 \\\because\forall^{n\in\mathbb N+1}_{\exists^{[k_0,k_1,k_2,k_3]\in\mathbb N^4}}\begin{bmatrix}2n-0\mapsto1n-0 \\2n-1\mapsto3n-1\end{bmatrix}^{k_0+k_1}_n&=&(3^{1+k_3}2^{1+k_2}-1) \\\because\forall^{n\in\mathbb N+1}_{\exists^{[k_0,k_1,k_2,k_3]\in\mathbb N^4}}\begin{bmatrix}2n-0\mapsto1n-0 \\2n-1\mapsto3n-1\end{bmatrix}^{k_0+0}_n&=&(3^{1+k_3}2^{1+k_2}-1)(1+2)^{k_1} \\\because\forall^{n\in\mathbb N+1}_{\exists^{[k_0,k_1,k_2,k_3]\in\mathbb N^4}}\begin{bmatrix}2n-0\mapsto1n-0 \\2n-1\mapsto3n-1\end{bmatrix}^{0}_n&=&(3^{1+k_3}2^{1+k_2}-1)3^{k_1}2^{k_0} \end{eqnarray}$$

Lerch Transcendent

$\sum_{n=N}^\infty R^n =\frac{1-R}{1-R}\sum_{n=N}^\infty R^n =\frac{\sum_{n=N}^\infty R^n-R\sum_{n=N}^\infty R^n}{1-R} =\frac{\sum_{n=N}^\infty R^n-\sum_{n=N}^\infty R^{n+1}}{1-R} =\frac{\sum_{n=N}^\infty R^n-\sum_{n=N+1}^\infty R^n}{1-R} =\frac{R^N}{1-R}$

$\sum_{n=N}^\infty nR^n =\frac{\sum_{n=N}^\infty nR^n-\sum_{n=N+1}^\infty[n-1]R^n}{1-R} =\frac{NR^N+\sum_{n=N+1}^\infty 1R^n}{1-R} =\frac{NR^N+\frac{R^{N+1}}{1-R}}{1-R} =\frac{R^N}{1-R}\left[N+\frac{R}{1-R}\right]$

$\sum_{n=N}^\infty n^2R^n =\frac{\sum_{n=N}^\infty n^2R^n-\sum_{n=N+1}^\infty[n-1]^2R^n}{1-R} =\frac{\sum_{n=N}^\infty n^2R^n-\sum_{n=N+1}^\infty n^2R^n+2\sum_{n=N+1}^\infty nR^n-\sum_{n=N+1}^\infty R^n}{1-R} =\frac{N^2R^N+2\frac{R^{N+1}}{1-R}\left[N+1+\frac{R}{1-R}\right]-\frac{R^{N+1}}{1-R}}{1-R} =\frac{R^N}{1-R}\left[N^2+\frac{R}{1-R}\left[2N+1+\frac{2R}{1-R}\right]\right]$

$\sum_{n=N}^\infty n^3R^n =\frac{N^3R^N+3\frac{R^{N+1}}{1-R}\left[N^2+2N+1+\frac{R}{1-R}\left[2N+3+\frac{2R}{1-R}\right]\right]-3\frac{R^{N+1}}{1-R}\left[N+1+\frac{R}{1-R}\right]+\frac{R^{N+1}}{1-R}}{1-R} =\frac{R^N}{1-R}\left[N^3+\frac{R}{1-R}\left[3N^2+3N+1+\frac{R}{1-R}\left[6N+6+\frac{6R}{1-R}\right]\right]\right]$

$\sum_{n=N}^\infty n^4R^n =\frac{R^N}{1-R}\left[N^4+\frac{R}{1-R}\left[4N^3+6N^2+4N+1+\frac{R}{1-R}\left[12N^2+24N+14+\frac{R}{1-R}\left[24N+36+\frac{24R}{1-R}\right]\right]\right]\right]$

apparently $\sum_{n=N}^\infty n^pR^n=\frac{R^N}{1-R}\sum_{r=0}^p\frac{\sum_{n=0}^{p-r}Ϥ(p,r,n)N^n}{(R^{-1}-1)^r}$ where $Ϥ(p,r,n)=[r=0][p=n]-[r\ne0]\sum_{s=1}^p(-1)^s{p\choose s}\sum_{m=n}^{p-r}{m\choose n}Ϥ(p-s,r-1,m)$

import Data.Function.Memoize

c::Int->Int->Int
c = memoFix2$ \c n k->
   if k<0||k>n then 0 
   else if k==0 then 1
   else c(n-1)(k-1)+c(n-1)k

q::Int->Int->Int->Int q = memoFix3$ \q p r n->if r==0 then sum[1|p==n] else sum[

 -(-1)^s*c p s*sum[c m n*q(p-s)(r-1)m|m<-[n..p-r]]|s<-[1..p]]

main=mapM_ print[[[q p r n|n<-[0..p-r]]|r<-[0..p]]|p<-[0..9]]

$\sum_{n=N}^\infty n^5R^n =\frac{R^N}{1-R}\left[N^5+\frac{R}{1-R}\left[5N^4+10N^3+10N^2+5N+1+\frac{R}{1-R}\left[20N^3+60N^2+70N+30+\frac{R}{1-R}\left[60N^2+180N+150+\frac{R}{1-R}\left[120N+240+\frac{120R}{1-R}\right]\right]\right]\right]\right]$

$\sum_{n=N}^\infty n^6R^n =\frac{R^N}{1-R}\begin{bmatrix} (R^{-1}-1)^{-0}\\ (R^{-1}-1)^{-1}\\ (R^{-1}-1)^{-2}\\ (R^{-1}-1)^{-3}\\ (R^{-1}-1)^{-4}\\ (R^{-1}-1)^{-5}\\ (R^{-1}-1)^{-6}\\ \end{bmatrix}^T\begin{bmatrix} 0&0&0&0&0&0&1\\ 1&6&15&20&15&6&0\\ 62&180&210&120&30&0&0\\ 540&900&540&120&0&0&0\\ 1560&1440&360&0&0&0&0\\ 1800&720&0&0&0&0&0\\ 720&0&0&0&0&0&0\end{bmatrix}\begin{bmatrix} 1\\ N\\ N^2\\ N^3\\ N^4\\ N^5\\ N^6\end{bmatrix}$

Arithmetico-geometric summations

$\sum_{n=N}^\infty [K+Ln]R^n \\=\frac{R^N}{1-R}\begin{bmatrix}(R^{-1}-1)^{-0}\\(R^{-1}-1)^{-1}\end{bmatrix}^T\begin{bmatrix}K&L\\L&0\end{bmatrix}\begin{bmatrix}N^0\\N^1\end{bmatrix}$

$\sum_{n=N}^\infty [K+Ln]R^n\sum_{n_2=n+1}^\infty [K+Ln_2]R^{n_2}$

$=\sum_{n=N}^\infty [K+Ln]\frac{R^{2n+1}}{1-R}\begin{bmatrix}(R^{-1}-1)^{-0}\\(R^{-1}-1)^{-1}\end{bmatrix}^T\begin{bmatrix}K+L&L\\L&0\end{bmatrix}\begin{bmatrix}n^0\\n^1\end{bmatrix}$

$=\sum_{n=N}^\infty\frac{R^{2n+1}}{1-R}\begin{bmatrix}(R^{-1}-1)^{-0}\\(R^{-1}-1)^{-1}\end{bmatrix}^T\begin{bmatrix}K^2+KL&2KL+L^2&L^2\\KL&L^2&0\end{bmatrix}\begin{bmatrix}n^0\\n^1\\n^2\end{bmatrix}$

$=\frac{R^{2N+1}}{1-R}\begin{bmatrix} (R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-2}(R^{-1}-1)^{-1} \end{bmatrix}^T\begin{bmatrix} K^2+KL&2KL+L^2&L^2\\KL&L^2&0 \\2KL+2L^2&2L^2&0\\L^2&0&0 \\2L^2&0&0\\0&0&0\end{bmatrix}\begin{bmatrix}N^0\\N^1\\N^2\end{bmatrix}$

$\sum_{n=N}^\infty [K+Ln]R^n\sum_{n_2=n+1}^\infty [K+Ln_2]R^{n_2}\sum_{n_3=n_2+1}^\infty [K+Ln_3]R^{n_3}$

$=\sum_{n=N}^\infty[K+Ln]\frac{R^{3n+3}}{1-R}\begin{bmatrix} (R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\end{bmatrix}^T\begin{bmatrix} K^2+3KL+2L^2&2KL+3L^2&L^2\\KL+L^2&L^2&0 \\2KL+4L^2&2L^2&0\\L^2&0&0 \\2L^2&0&0\\0&0&0\end{bmatrix}\begin{bmatrix}n^0\\n^1\\n^2\end{bmatrix}$

$=\sum_{n=N}^\infty\frac{R^{3n+3}}{1-R}\begin{bmatrix} (R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\end{bmatrix}^T\begin{bmatrix} K^3+3K^2L+2KL^2&3K^2L+6KL^2+2L^3&3KL^2+3L^3&L^3 \\K^2L+KL^2&2KL^2+L^3&L^3&0 \\2K^2L+4KL^2&4KL^2+4L^3&2L^3&0 \\KL^2&L^3&0&0 \\2KL^2&2L^3&0&0 \\0&0&0&0\end{bmatrix} \begin{bmatrix}n^0\\n^1\\n^2\\n^3\end{bmatrix}$

$=\frac{R^{3N+3}}{1-R}\begin{bmatrix} (R^{-3}-1)^{-0}(R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\\ \end{bmatrix}^T\begin{bmatrix} K^3+3K^2L+2KL^2&3K^2L+6KL^2+2L^3&3KL^2+3L^3&L^3 \\K^2L+KL^2&2KL^2+L^3&L^3&0 \\2K^2L+4KL^2&4KL^2+4L^3&2L^3&0 \\KL^2&L^3&0&0 \\2KL^2&2L^3&0&0 \\0&0&0&0 %\\\begin{bmatrix}1&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix} %&\begin{bmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix} %&\begin{bmatrix}0&0&1&0\\1&2&0&0\\2&0&0&0\\0&0&0&0\end{bmatrix} %&\begin{bmatrix}0&0&0&1\\1&3&3&0\\6&6&0&0\\6&0&0&0\end{bmatrix} \\3K^2L+9KL^2+6L^3&6KL^2+9L^3&3L^3&0 \\2KL^2+2L^3&2L^3&0&0 \\4KL^2+6L^3&4L^3&0&0 \\L^3&0&0&0 \\2L^3&0&0&0 \\0&0&0&0 \\6KL^2+12L^3&6L^3&0&0 \\2L^3&0&0&0 \\4L^3&0&0&0 \\0&0&0&0 \\0&0&0&0 \\0&0&0&0 \\6L^3&0&0&0 \\0&0&0&0 \\0&0&0&0 \\0&0&0&0 \\0&0&0&0 \\0&0&0&0 \end{bmatrix} \begin{bmatrix}N^0\\N^1\\N^2\\N^3\end{bmatrix}$

Base N=0

$\sum_{n=0}^\infty(K+Ln)R^n =\frac{1}{1-R}\begin{bmatrix}(R^{-1}-1)^{-0}\\(R^{-1}-1)^{-1}\end{bmatrix}^T\begin{bmatrix}K\\L\end{bmatrix} \\=\frac{K+L(R^{-1}-1)^{-1}}{1-R}\overset!=1 \Longrightarrow K=(1-R)-L(R^{-1}-1)^{-1}$

$\sum_{n=0}^\infty(K+Ln)R^n\sum_{n_2=n+1}^\infty(K+Ln_2)R^{n_2} \\=\frac{R}{1-R}\begin{bmatrix} (R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\(R^{-2}-1)^{-2}(R^{-1}-1)^{-1} \end{bmatrix}^T\begin{bmatrix} K^2+KL\\KL\\2KL+2L^2\\L^2\\2L^2\\0\end{bmatrix}=\begin{bmatrix} (R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\(R^{-2}-1)^{-0}(R^{-1}-1)^{-2}\\ (R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\(R^{-2}-1)^{-1}(R^{-1}-1)^{-2}\\ (R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\\(R^{-2}-1)^{-2}(R^{-1}-1)^{-2} \end{bmatrix}^T\begin{bmatrix} K^2+KL\\KL\\2KL+2L^2\\L^2\\2L^2\\0\end{bmatrix} \\=\begin{bmatrix} (R^{-1}-1)^{-1}\\ (R^{-1}-1)^{-1}+(R^{-1}-1)^{-2}+2(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ 2(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}+(R^{-2}-1)^{-1}(R^{-1}-1)^{-2}+2(R^{-2}-1)^{-2}(R^{-1}-1)^{-1} \end{bmatrix}^T\begin{bmatrix} K^2\\KL\\L^2\end{bmatrix} \\\overset!=\begin{bmatrix} (R^{-1}-1)^{-1}\\ (R^{-1}-1)^{-1}(1+(R^{-1}-1)^{-1}+2(R^{-2}-1)^{-1})\\ (R^{-1}-1)^{-1}(R^{-2}-1)^{-1}(2+(R^{-1}-1)^{-1}+2(R^{-2}-1)^{-1}) \end{bmatrix}^T\begin{bmatrix} (1-R)^2-2L(1-R)(R^{-1}-1)^{-1}+L^2(R^{-1}-1)^{-2}\\ L(1-R)-L^2(R^{-1}-1)^{-1}\\L^2\end{bmatrix} \\=\begin{bmatrix}(1-R)^2(R^{-1}-1)^{-1} \\(1-R)(R^{-1}-1)^{-1}(1+(R^{-1}-1)^{-1}+2(R^{-2}-1)^{-1})-2(1-R)(R^{-1}-1)^{-2} \\(R^{-1}-1)^{-1}(R^{-2}-1)^{-1}(2+(R^{-1}-1)^{-1}+2(R^{-2}-1)^{-1})-(R^{-1}-1)^{-2}(1+(R^{-1}-1)^{-1}+2(R^{-2}-1)^{-1})+(R^{-1}-1)^{-3} \end{bmatrix}^T\begin{bmatrix}1\\L\\L^2\end{bmatrix} \\=\begin{bmatrix}[1-R]R\\R/[1+R]\\ \left[2\frac{R^2}{1-R^2}-\frac{R}{1-R}\right] \left[1+\frac{R^2}{1-R^2}\right]\frac{R}{1-R} \end{bmatrix}^T\begin{bmatrix}1\\L\\L^2\end{bmatrix} =\begin{bmatrix}[1-R]R\\R/[1+R]\\ \left[\frac{2R^2-R(1+R)}{1-R^2}\right] \left[\frac{1}{1-R^2}\right]\frac{R}{1-R} \end{bmatrix}^T\begin{bmatrix}1\\L\\L^2\end{bmatrix} \\=\begin{bmatrix}R-R^2\\R/[1+R]\\-[R/[1-R^2]]^2 \end{bmatrix}^T\begin{bmatrix}1\\L\\L^2\end{bmatrix}\overset!=\frac12 \Longrightarrow L=\frac{-b\pm\sqrt{b^2-4ac}}{2a} =\frac{-\frac{R}{1+R}\pm\sqrt{(\frac{R}{1+R})^2+4(\frac{R}{1-R^2})^2(R-R^2-\frac12)}} {-2(\frac{R}{1-R^2})^2} =\\=\frac{\frac{R}{1+R}\pm(\frac{R}{1-R^2})\sqrt{(1-2R+R^2)+(4R-4R^2-2)}} {2(\frac{R}{1-R^2})^2} =\frac{(\frac{R}{1-R^2})(1-R\pm\sqrt{2R-1-3R^2})}{2(\frac{R}{1-R^2})^2} =\frac{1-R^2}{2R}(1-R\pm\sqrt{2R-1-3R^2}) \\ \Longrightarrow\begin{bmatrix}K\\L\end{bmatrix} =\begin{bmatrix}1-R+\frac12(R+1)(R-1\pm\sqrt{2R-1-3R^2})\\ \frac12(R-\frac1R)(R-1\pm\sqrt{2R-1-3R^2})\end{bmatrix} =\begin{bmatrix}\frac{(1-2R+R^2)}2\pm\frac{(1+R)\sqrt{2R-1-3R^2}}2\\ \frac{(1-R-R^2+R^3)}{2R}\pm\frac{(1-R^2)\sqrt{2R-1-3R^2}}{2R}\end{bmatrix} \\ \Longrightarrow\begin{bmatrix}K^2\\L^2\end{bmatrix} =\begin{bmatrix} \frac{(6R^2-1-4R-12R^3-5R^4)}4 \pm\frac{(2-2R-2R^2+2R^3)\sqrt{2R-1-3R^2}}4 \\ \frac{(-2R^2+4R^4-2R^6)}{4R^2} \pm\frac{(2-2R-4R^2+4R^3+2R^4-2R^5)\sqrt{2R-1-3R^2}}{4R^2}\end{bmatrix} $

$\sum_{n=0}^\infty(K+Ln)R^n \sum_{n_2=n+1}^\infty(K+Ln_2)R^{n_2} \sum_{n_3=n_2+1}^\infty(K+Ln_3)R^{n_3} \\=\frac{R^3}{1-R}\begin{bmatrix} (R^{-3}-1)^{-0}(R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-0}(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-1}(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-2}(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-0}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-0}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-1}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-1}(R^{-1}-1)^{-1}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-2}(R^{-1}-1)^{-0}\\ (R^{-3}-1)^{-3}(R^{-2}-1)^{-2}(R^{-1}-1)^{-1}\\ \end{bmatrix}^T\begin{bmatrix} K^3+3K^2L+2KL^2 \\K^2L+KL^2 \\2K^2L+4KL^2 \\KL^2 \\2KL^2 \\0 \\3K^2L+9KL^2+6L^3 \\2KL^2+2L^3 \\4KL^2+6L^3 \\L^3 \\2L^3 \\0 \\6KL^2+12L^3 \\2L^3 \\4L^3 \\0 \\0 \\0 \\6L^3 \\0 \\0 \\0 \\0 \\0\end{bmatrix} \\=\frac{R^3}{1-R}\begin{bmatrix} 1\\ 3 +\frac{R}{1-R} +2\frac{R^2}{1-R^2} +3\frac{R^3}{1-R^3}\\ 2 +\frac{R}{1-R} +4\frac{R^2}{1-R^2} +\frac{R}{1-R}\frac{R^2}{1-R^2} +2\frac{R^2}{1-R^2}\frac{R^2}{1-R^2} +9\frac{R^3}{1-R^3} +2\frac{R^3}{1-R^3}\frac{R}{1-R} +4\frac{R^3}{1-R^3}\frac{R^2}{1-R^2} +6\frac{R^3}{1-R^3}\frac{R^3}{1-R^3} \\ 6\frac{R^3}{1-R^3} +2\frac{R^3}{1-R^3}\frac{R}{1-R} +6\frac{R^3}{1-R^3}\frac{R^2}{1-R^2} +\frac{R^3}{1-R^3}\frac{R^2}{1-R^2}\frac{R}{1-R} +2\frac{R^3}{1-R^3}\frac{R^2}{1-R^2}\frac{R^2}{1-R^2} +12\frac{R^3}{1-R^3}\frac{R^3}{1-R^3} +2\frac{R^3}{1-R^3}\frac{R^3}{1-R^3}\frac{R}{1-R} +4\frac{R^3}{1-R^3}\frac{R^3}{1-R^3}\frac{R^2}{1-R^2} +6\frac{R^3}{1-R^3}\frac{R^3}{1-R^3}\frac{R^3}{1-R^3} \end{bmatrix}^T\begin{bmatrix} K^3\\ K^2L\\ KL^2\\ L^3\end{bmatrix} \\=\frac{R^3}{1-R}\begin{bmatrix} 1 \\ \frac{ 3(1-R)(1+R)(1+R+R^2) +R(1+R)(1+R+R^2) +2R^2(1+R+R^2) +3R^3(1+R)}{(1-R)(1+R)(1+R+R^2)} \\ \frac{2(1-R)^2(1+R)^2(1+R+R^2)^2 +R(1-R)(1+R)^2(1+R+R^2)^2 +4R^2(1-R)(1+R)(1+R+R^2)^2 +R^3(1+R)(1+R+R^2)^2 +2R^4(1+R+R^2)^2 +9R^3(1-R)(1+R)^2(1+R+R^2) +2R^4(1+R)^2(1+R+R^2) +4R^5(1+R)(1+R+R^2) +6R^6(1+R)^2 }{(1-R)^2(1+R)^2(1+R+R^2)^2} \\ \frac{ 6R^3(1-R)^2(1+R)^3(1+R+R^2)^2 +2R^4(1-R)(1+R)^3(1+R+R^2)^2 +6R^5(1-R)(1+R)^2(1+R+R^2)^2 +R^6(1+R)^2(1+R+R^2)^2 +2R^7(1+R)(1+R+R^2)^2 +12R^6(1-R)(1+R)^3(1+R+R^2) +2R^7(1+R)^3(1+R+R^2) +4R^8(1+R)^2(1+R+R^2) +6R^9(1+R)^3 }{(1-R)^3(1+R)^3(1+R+R^2)^3} \end{bmatrix}^T\begin{bmatrix} K^3\\ K^2L\\ KL^2\\ L^3\end{bmatrix} \\=\frac{R^3}{1-R}\begin{bmatrix} 1\\ \frac{3+4R+4R^2+4R^3+3R^4 }{(1-R)(1+R)(1+R+R^2)}\\ \frac{2+5R+9R^2+18R^3+27R^4+22R^5+14R^6+8R^7+3R^8 }{(1-R)^2(1+R)^2(1+R+R^2)^2}\\ \frac{6R^3+20R^4+32R^5+39R^6+44R^7+38R^8+22R^9+46R^{10}-32R^{11}+R^{12} }{(1-R)^3(1+R)^3(1+R+R^2)^3} \end{bmatrix}^T\begin{bmatrix} (\frac{(1-2R+R^2)}2\pm\frac{(1+R)\sqrt{2R-1-3R^2}}2) (\frac{(6R^2-1-4R-12R^3-5R^4)}4\pm\frac{(2-2R-2R^2+2R^3)\sqrt{2R-1-3R^2}}4) \\ (\frac{(1-R-R^2+R^3)}{2R}\pm\frac{(1-R^2)\sqrt{2R-1-3R^2}}{2R}) (\frac{(6R^2-1-4R-12R^3-5R^4)}4\pm\frac{(2-2R-2R^2+2R^3)\sqrt{2R-1-3R^2}}4) \\ (\frac{(1-2R+R^2)}2\pm\frac{(1+R)\sqrt{2R-1-3R^2}}2) (\frac{(-2R^2+4R^4-2R^6)}{4R^2}\pm\frac{(2-2R-4R^2+4R^3+2R^4-2R^5)\sqrt{2R-1-3R^2}}{4R^2}) \\ (\frac{(1-R-R^2+R^3)}{2R}\pm\frac{(1-R^2)\sqrt{2R-1-3R^2}}{2R}) (\frac{(-2R^2+4R^4-2R^6)}{4R^2}\pm\frac{(2-2R-4R^2+4R^3+2R^4-2R^5)\sqrt{2R-1-3R^2}}{4R^2}) \end{bmatrix} $

Exact Covers and Hittings

$c(s) = \{ r\subseteq s : \forall x \in \cup s : \exists! y \in r: y \ni x \}$

$h(s) = \{ r\subseteq \cup s: \forall x \in s : \exists! y \in r: y \in x \}$

exacthittings sets = if null sets then [[]] else [ pivot : extension |
 pivot <- minimumBy (comparing length) sets,
 let (deletables, undeletables) = partition (elem pivot) sets,
 extension <- exacthittings (map (\\ unions deletables) undeletables) ]
test0 = exacthittings (words "ab ef de abc cd de acef") == words "bdf"

sudokus = exacthittings (concat [ [ 
      row x >< [y],
      col x >< [y],
      [x] >< [y] >< [0..8],
      box (divMod x 3) >< [y] ] | (x,y) <- [0..8] >< [0..8] ])
              where
                      (><) = liftM2 (,)
                      row i = [i] >< [0..8]
                      col i = [0..8] >< [i]
                      box (i,j) = map (3*i+) [0..2] >< map (3*j+) [0..2]
test1 = sudokus

Sedenion Multiplication

tee --append PowerNum.lhs <<<EOF
partner row col = case compare row col of
  GT -> 0 - partner (umbox - row) col where umbox = until (>row) (2*) 1
  LT -> 0 - flip partner row col
  EQ -> row
EOF

$ \begin{bmatrix} +P0&-P1&-P2&-P3&-P4&-P5&-P6&-P7&-P8&-P9&-PA&-PB&-PC&-PD&-PE&-PF \\ +P1&+P0&-P3&+P2&-P5&+P4&+P7&-P6&-P9&+P8&+PB&-PA&+PD&-PC&-PF&+PE \\ +P2&+P3&+P0&-P1&-P6&-P7&+P4&+P5&-PA&-PB&+P8&+P9&+PE&+PF&-PC&-PD \\ +P3&-P2&+P1&+P0&-P7&+P6&-P5&+P4&-PB&+PA&-P9&+P8&+PF&-PE&+PD&-PC \\ +P4&+P5&+P6&+P7&+P0&-P1&-P2&-P3&-PC&-PD&-PE&-PF&+P8&+P9&+PA&+PB \\ +P5&-P4&+P7&-P6&+P1&+P0&+P3&-P2&-PD&+PC&-PF&+PE&-P9&+P8&-PB&+PA \\ +P6&-P7&-P4&+P5&+P2&-P3&+P0&+P1&-PE&+PF&+PC&-PD&-PA&+PB&+P8&-P9 \\ +P7&+P6&-P5&-P4&+P3&+P2&-P1&+P0&-PF&-PE&+PD&+PC&-PB&-PA&+P9&+P8 \\ +P8&+P9&+PA&+PB&+PC&+PD&+PE&+PF&+P0&-P1&-P2&-P3&-P4&-P5&-P6&-P7 \\ +P9&-P8&+PB&-PA&+PD&-PC&-PF&+PE&+P1&+P0&+P3&-P2&+P5&-P4&-P7&+P6 \\ +PA&-PB&-P8&+P9&+PE&+PF&-PC&-PD&+P2&-P3&+P0&+P1&+P6&+P7&-P4&-P5 \\ +PB&+PA&-P9&-P8&+PF&-PE&+PD&-PC&+P3&+P2&-P1&+P0&+P7&-P6&+P5&-P4 \\ +PC&-PD&-PE&-PF&-P8&+P9&+PA&+PB&+P4&-P5&-P6&-P7&+P0&+P1&+P2&+P3 \\ +PD&+PC&-PF&+PE&-P9&-P8&-PB&+PA&+P5&+P4&-P7&+P6&-P1&+P0&-P3&+P2 \\ +PE&+PF&+PC&-PD&-PA&+PB&-P8&-P9&+P6&+P7&+P4&-P5&-P2&+P3&+P0&-P1 \\ +PF&-PE&+PD&+PC&-PB&-PA&+P9&-P8&+P7&-P6&+P5&+P4&-P3&-P2&+P1&+P0 \\ \end{bmatrix} \begin{bmatrix} Q0\\Q1\\Q2\\Q3\\Q4\\Q5\\Q6\\Q7\\Q8\\Q9\\QA\\QB\\QC\\QD\\QE\\QF \end{bmatrix} =: \begin{bmatrix} P0\\P1\\P2\\P3\\P4\\P5\\P6\\P7\\P8\\P9\\PA\\PB\\PC\\PD\\PE\\PF \end{bmatrix} \begin{bmatrix} Q0\\Q1\\Q2\\Q3\\Q4\\Q5\\Q6\\Q7\\Q8\\Q9\\QA\\QB\\QC\\QD\\QE\\QF \end{bmatrix}$

sudo apt-get install open-axiom ; open-axiom
octon(p0,p1,p2,p3,p4,p5,p6,p7)*octon(q0,q1,q2,q3,q4,q5,q6,q7)-conjugate(octon(q8,q9,qa,qb,qc,qd,qe,qf))*octon(p8,p9,pa,pb,pc,pd,pe,pf)
octon(q8,q9,qa,qb,qc,qd,qe,qf)*octon(p0,p1,p2,p3,p4,p5,p6,p7)+octon(p8,p9,pa,pb,pc,pd,pe,pf)*conjugate(octon(q0,q1,q2,q3,q4,q5,q6,q7))
)quit