0132_0231_0312_1023_2103
Counting sequence:
1, 1, 2, 6, 19, 53, 147, 413, 1166, 3287, 9260, 26092, 73534, 207242, 584065, 1646057, 4639070, 13074285, 36847250, 103846592, 292670864, 824834427, 2324631328, 6551509864, 18464124329, 52037453379, 146657188462, 413324049427, 1164871437413, 3282957930962, 9252362477164, 26075939203567, 73489836466349, 207116454053460, 583716437575112, 1645088416824557, 4636353758375910, 13066638821958270, 36825716716540014, 103785941447186637, 292499986490219749, 824352902751502126, 2323274323630020977, 6547685541984110547, 18453346434665610631, 52007078295730128671, 146571582690130262076, 413082787118489833610, 1164191488430110063112, 3281041631357894066942, 9246961770198461847674, 26060718389642676859693, 73446939639470750258641, 206995557902501251896506, 583375715879950316609050, 1644128160657171953764973, 4633647467804784917882711, 13059011681493052884142209, 36804221141615072836733453, 103725360454386119913379832, 292329251038735469072067317, 823871718917265253800462192, 2321918202916165700967646450, 6543863586091661112318954880, 18442575013880682934399051820, 51976721193504999853934723481, 146486027249124686651035198015, 412841666163288247092952189828, 1163511937084759580666059582427, 3279126451357956506573788264778, 9241564217155181272036177645792, 26045506462379452766461767036100, 73404067854940589629798165286039, 206874732324190470788569460866878, 583035193073770774667913775817288, 1643168465010360020388269458154225, 4630942756937272066140287889349305, 13051388993090595522718905018389310, 36782738113917406392301570527674085, 103664814823410318594417088862123361, 292158615247456565614825876272373078, 823390815955381600675666613219874136, 2320562873784949235555977749686280599, 6540043861116940761034652126550597609, 18431809880492450530565175653729629996, 51946381811053801140191796728543866586, 146400521747769143886798591078372295281, 412600685953034367676204184711871197330, 1162832782400986247803533935276097361658, 3277212389269599480104751686428405818080, 9236169814722835837221189731875797024677, 26030303414485080359388300795367029180283, 73361221095141453361379177364573339243994, 206753977273095457129348228024128792124181, 582694869034489435401747244712831838951253, 1642209329548426957381775234415976939454525, 4628239624839168784770864564218268735205925, 13043770754134995119753975861752054648815450, 36761267626098719694455412212972133069599852, 103604304533583626840405903760134408026697048, 291988079058161254059748344349284607100488664
Generating function in Maple syntax:
-(2*x-1)*(x-1)^2/(x^8+x^7-5*x^6+5*x^5+x^4-6*x^3+8*x^2-5*x+1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x -1\right)^{2}}{x^{8}+x^{7}-5 x^{6}+5 x^{5}+x^{4}-6 x^{3}+8 x^{2}-5 x +1}
Generating function in sympy syntax:
(1 - 2*x)*(x - 1)**2/(x**8 + x**7 - 5*x**6 + 5*x**5 + x**4 - 6*x**3 + 8*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^8+x^7-5*x^6+5*x^5+x^4-6*x^3+8*x^2-5*x+1)*F(x)+(2*x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}+x^{7}-5 x^{6}+5 x^{5}+x^{4}-6 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-687452274/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+6)-687452274/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+6)-687452274/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+6)-687452274/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+6)-687452274/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+6)-687452274/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+6)-687452274/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 7)^(-n+6)-687452274/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 8)^(-n+6)-74753032/661387319*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 7)^(-n+5)-74753032/661387319*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 8)^(-n+5)-74753032/661387319*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+5)-74753032/661387319*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+5)-74753032/661387319*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+5)-74753032/661387319*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+5)-74753032/661387319*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+5)-74753032/661387319*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+5)+3386974034/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 8)^(-n+4)+3386974034/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+4)+3386974034/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+4)+3386974034/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+4)+3386974034/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+4)+3386974034/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+4)+3386974034/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+4)+3386974034/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 7)^(-n+4)-2642037241/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+3)-2642037241/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+3)-2642037241/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+3)-2642037241/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+3)-2642037241/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+3)-2642037241/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+3)-2642037241/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 7)^(-n+3)-2642037241/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 8)^(-n+3)-2042322409/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+2)-2042322409/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+2)-2042322409/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+2)-2042322409/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+2)-2042322409/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+2)-2042322409/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+2)-2042322409/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 7)^(-n+2)-2042322409/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 8)^(-n+2)+3774580614/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+1)+3774580614/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+1)+3774580614/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+1)+3774580614/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+1)+3774580614/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+1)+3774580614/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+1)+3774580614/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 7)^(-n+1)+3774580614/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 8)^(-n+1)+1536289587/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n-1)+1536289587/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n-1)+1536289587/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n-1)+1536289587/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n-1)+1536289587/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n-1)+1536289587/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n-1)+1536289587/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 7)^(-n-1)+1536289587/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 8)^(-n-1)-3211072330/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n)-3211072330/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n)-3211072330/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n)-3211072330/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n)-3211072330/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n)-3211072330/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n)-3211072330/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 7)^(-n)-3211072330/7275260509*RootOf(_Z^8+_Z^7-5*_Z^6+5*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{687452274 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{7275260509}-\frac{687452274 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{7275260509}-\frac{687452274 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{7275260509}-\frac{687452274 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{7275260509}-\frac{687452274 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{7275260509}-\frac{687452274 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{7275260509}-\frac{687452274 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{7275260509}-\frac{687452274 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{7275260509}-\frac{74753032 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{661387319}-\frac{74753032 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{661387319}-\frac{74753032 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{661387319}-\frac{74753032 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{661387319}-\frac{74753032 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{661387319}-\frac{74753032 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{661387319}-\frac{74753032 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{661387319}-\frac{74753032 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{661387319}+\frac{3386974034 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{7275260509}+\frac{3386974034 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{7275260509}+\frac{3386974034 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{7275260509}+\frac{3386974034 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{7275260509}+\frac{3386974034 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{7275260509}+\frac{3386974034 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{7275260509}+\frac{3386974034 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{7275260509}+\frac{3386974034 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{7275260509}-\frac{2642037241 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{7275260509}-\frac{2642037241 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{7275260509}-\frac{2642037241 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{7275260509}-\frac{2642037241 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{7275260509}-\frac{2642037241 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{7275260509}-\frac{2642037241 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{7275260509}-\frac{2642037241 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{7275260509}-\frac{2642037241 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{7275260509}-\frac{2042322409 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{7275260509}-\frac{2042322409 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{7275260509}-\frac{2042322409 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{7275260509}-\frac{2042322409 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{7275260509}-\frac{2042322409 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{7275260509}-\frac{2042322409 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{7275260509}-\frac{2042322409 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{7275260509}-\frac{2042322409 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{7275260509}+\frac{3774580614 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{7275260509}+\frac{3774580614 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{7275260509}+\frac{3774580614 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{7275260509}+\frac{3774580614 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{7275260509}+\frac{3774580614 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{7275260509}+\frac{3774580614 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{7275260509}+\frac{3774580614 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{7275260509}+\frac{3774580614 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{7275260509}+\frac{1536289587 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{7275260509}+\frac{1536289587 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{7275260509}+\frac{1536289587 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{7275260509}+\frac{1536289587 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{7275260509}+\frac{1536289587 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{7275260509}+\frac{1536289587 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{7275260509}+\frac{1536289587 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{7275260509}+\frac{1536289587 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{7275260509}-\frac{3211072330 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{7275260509}-\frac{3211072330 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{7275260509}-\frac{3211072330 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{7275260509}-\frac{3211072330 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{7275260509}-\frac{3211072330 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{7275260509}-\frac{3211072330 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{7275260509}-\frac{3211072330 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{7275260509}-\frac{3211072330 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-5 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{7275260509}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 147
a(7) = 413
a(n+8) = -a(n)-a(n+1)+5*a(n+2)-5*a(n+3)-a(n+4)+6*a(n+5)-8*a(n+6)+5*a(n+7), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 147
a \! \left(7\right) = 413
a \! \left(n +8\right) = -a \! \left(n \right)-a \! \left(n +1\right)+5 a \! \left(n +2\right)-5 a \! \left(n +3\right)-a \! \left(n +4\right)+6 a \! \left(n +5\right)-8 a \! \left(n +6\right)+5 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18586/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[20,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[53,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[35,x]
F[25,x] = F[14,x]+F[26,x]
F[26,x] = F[22,x]+F[27,x]+F[32,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[29,x]*F[4,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[14,x]+F[34,x]
F[34,x] = F[32,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[2,x]+F[36,x]
F[39,x] = 2*F[22,x]+F[40,x]+F[47,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]+F[45,x]
F[43,x] = F[44,x]
F[44,x] = F[2,x]*F[4,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[42,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[22,x]+F[23,x]+F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[2,x]+F[49,x]
F[52,x] = F[47,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[66,x]
F[55,x] = F[2,x]+F[56,x]
F[56,x] = F[22,x]+F[57,x]+F[65,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[4,x]+F[60,x]
F[60,x] = F[22,x]+F[27,x]+F[61,x]
F[61,x] = F[14,x]*F[4,x]
F[62,x] = F[43,x]+F[63,x]
F[63,x] = 2*F[22,x]+F[40,x]+F[64,x]
F[64,x] = F[4,x]*F[49,x]
F[65,x] = F[4,x]*F[55,x]
F[66,x] = F[49,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[43,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{20}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{53}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{25}\! \left(x \right) = F_{14}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{40}\! \left(x \right)+F_{47}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{47}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{66}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{22}\! \left(x \right)+F_{57}\! \left(x \right)+F_{65}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{62}\! \left(x \right) = F_{43}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{40}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{66}\! \left(x \right) = F_{49}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{43}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_20(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_53(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_35(x))
Eq(F_25(x), F_14(x) + F_26(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_32(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_29(x)*F_4(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_14(x) + F_34(x))
Eq(F_34(x), F_32(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_2(x) + F_36(x))
Eq(F_39(x), 2*F_22(x) + F_40(x) + F_47(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x) + F_45(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_2(x)*F_4(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_42(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_22(x) + F_23(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_2(x) + F_49(x))
Eq(F_52(x), F_47(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_66(x))
Eq(F_55(x), F_2(x) + F_56(x))
Eq(F_56(x), F_22(x) + F_57(x) + F_65(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), F_4(x) + F_60(x))
Eq(F_60(x), F_22(x) + F_27(x) + F_61(x))
Eq(F_61(x), F_14(x)*F_4(x))
Eq(F_62(x), F_43(x) + F_63(x))
Eq(F_63(x), 2*F_22(x) + F_40(x) + F_64(x))
Eq(F_64(x), F_4(x)*F_49(x))
Eq(F_65(x), F_4(x)*F_55(x))
Eq(F_66(x), F_49(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_43(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 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