0123_0231_0321_1032_2103
Counting sequence:
1, 1, 2, 6, 19, 51, 132, 352, 954, 2575, 6916, 18574, 49943, 134331, 361222, 971216, 2611387, 7021722, 18880629, 50767423, 136506307, 367046539, 986938288, 2653742360, 7135548797, 19186510677, 51589895959, 138718157898, 372994104528, 1002929991075, 2696741199547, 7251167260098, 19497394354237, 52425819559621, 140965839133334, 379037809457187, 1019180688675420, 2740437102149677, 7368659545982723, 19813314986125030, 53275286813232066, 143249940104591518, 385179443742639214, 1035694700978480838, 2784841016468501356, 7488055582093472077, 20134354553433750680, 54138518182583458628, 145571050873643937100, 391420592284965357219, 1052476293502205285588, 2829964417349042792542, 7609386218868706797474, 20460595996521099369010, 55015736682517120850247, 147929771118822402330143, 397762867550971492372841, 1069529801917205593232558, 2875818962770506113701133, 7732682803046055370430555, 20792123602564847972770291, 55907168949699824604360878, 150326710234865813765631620, 404207908119422732523478569, 1086859632137319561306404172, 2922416499631339809753443121, 7857977189310531659641564950, 21129023024443665974557230992, 56813045292975883252246013074, 152762487490673560554010127691, 410757379119464378674435003612, 1104470261466462217458427254747, 2969769066787754040502818599949, 7985301748495486161534473726685, 21471381302889125437251404264798, 57733599752935788250520171927093, 155237732189297296361260050603641, 417412972660700129941055194462060, 1122366239755303172057802943958573, 3017888898164236006206996616075641, 8114689375946112913753621196171892, 21819286888973207954900353836308192, 58669070162381978640520668974686341, 157753083830526552150460974790806786, 424176408270363471798588415337059166, 1140552190576758479766729774245093116, 3066788425914283353139162172906193142, 8246173500018230067339090172561156248, 22172829666960366703450681574590482566, 59619698207774902517983416602096391004, 160309192276108757117377830650355430815, 431049433337571646191704699188828805531, 1159032812420528656858836705981626574065, 3116480283632353108508650623324083501949, 8379788090714770073284043417449321524020, 22532100977529864096537006634908774250538, 60585729491674698288828282337502239665332, 162906717917646356471953527161578824505478, 438033823564777976019337610949990464605598, 1177812879906992069765237972045001368489757, 3166977309617854529553584734946827643121340
Generating function in Maple syntax:
-(2*x^3+x^2-1)/(3*x^8+4*x^7-2*x^6-7*x^5-5*x^4-4*x^3-2*x^2-x+1)
Generating function in latex syntax:
-\frac{2 x^{3}+x^{2}-1}{3 x^{8}+4 x^{7}-2 x^{6}-7 x^{5}-5 x^{4}-4 x^{3}-2 x^{2}-x +1}
Generating function in sympy syntax:
(-2*x**3 - x**2 + 1)/(3*x**8 + 4*x**7 - 2*x**6 - 7*x**5 - 5*x**4 - 4*x**3 - 2*x**2 - x + 1)
Implicit equation for the generating function in Maple syntax:
(3*x^8+4*x^7-2*x^6-7*x^5-5*x^4-4*x^3-2*x^2-x+1)*F(x)+2*x^3+x^2-1 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x^{8}+4 x^{7}-2 x^{6}-7 x^{5}-5 x^{4}-4 x^{3}-2 x^{2}-x +1\right) F \! \left(x \right)+2 x^{3}+x^{2}-1 = 0
Explicit closed form in Maple syntax:
-122018733282/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 1)^(-n+6)-122018733282/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 2)^(-n+6)-122018733282/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 3)^(-n+6)-122018733282/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 4)^(-n+6)-122018733282/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 5)^(-n+6)-122018733282/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 6)^(-n+6)-122018733282/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 7)^(-n+6)-122018733282/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 8)^(-n+6)-18861399631/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 1)^(-n+5)-18861399631/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 2)^(-n+5)-18861399631/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 3)^(-n+5)-18861399631/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 4)^(-n+5)-18861399631/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 5)^(-n+5)-18861399631/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 6)^(-n+5)-18861399631/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 7)^(-n+5)-18861399631/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 8)^(-n+5)+42387119144/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 1)^(-n+4)+42387119144/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 2)^(-n+4)+42387119144/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 3)^(-n+4)+42387119144/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 4)^(-n+4)+42387119144/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 5)^(-n+4)+42387119144/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 6)^(-n+4)+42387119144/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 7)^(-n+4)+42387119144/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 8)^(-n+4)-269010293109/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 1)^(-n+3)-269010293109/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 2)^(-n+3)-269010293109/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 3)^(-n+3)-269010293109/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 4)^(-n+3)-269010293109/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 5)^(-n+3)-269010293109/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 6)^(-n+3)-269010293109/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 7)^(-n+3)-269010293109/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 8)^(-n+3)+13102182087/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 1)^(-n+2)+13102182087/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 2)^(-n+2)+13102182087/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 3)^(-n+2)+13102182087/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 4)^(-n+2)+13102182087/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 5)^(-n+2)+13102182087/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 6)^(-n+2)+13102182087/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 7)^(-n+2)+13102182087/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 8)^(-n+2)+566392852412/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 1)^(-n+1)+566392852412/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 2)^(-n+1)+566392852412/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 3)^(-n+1)+566392852412/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 4)^(-n+1)+566392852412/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 5)^(-n+1)+566392852412/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 6)^(-n+1)+566392852412/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 7)^(-n+1)+566392852412/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 8)^(-n+1)+250319183702/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 1)^(-n-1)+250319183702/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 2)^(-n-1)+250319183702/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 3)^(-n-1)+250319183702/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 4)^(-n-1)+250319183702/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 5)^(-n-1)+250319183702/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 6)^(-n-1)+250319183702/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 7)^(-n-1)+250319183702/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 8)^(-n-1)+733632917937/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 1)^(-n)+733632917937/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 2)^(-n)+733632917937/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 3)^(-n)+733632917937/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 4)^(-n)+733632917937/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 5)^(-n)+733632917937/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 6)^(-n)+733632917937/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 7)^(-n)+733632917937/4590701992207*RootOf(3*_Z^8+4*_Z^7-2*_Z^6-7*_Z^5-5*_Z^4-4*_Z^3-2*_Z^2-_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{122018733282 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{4590701992207}-\frac{122018733282 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{4590701992207}-\frac{122018733282 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{4590701992207}-\frac{122018733282 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{4590701992207}-\frac{122018733282 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{4590701992207}-\frac{122018733282 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{4590701992207}-\frac{122018733282 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{4590701992207}-\frac{122018733282 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{4590701992207}-\frac{18861399631 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{4590701992207}-\frac{18861399631 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{4590701992207}-\frac{18861399631 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{4590701992207}-\frac{18861399631 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{4590701992207}-\frac{18861399631 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{4590701992207}-\frac{18861399631 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{4590701992207}-\frac{18861399631 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{4590701992207}-\frac{18861399631 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{4590701992207}+\frac{42387119144 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{4590701992207}+\frac{42387119144 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{4590701992207}+\frac{42387119144 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{4590701992207}+\frac{42387119144 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{4590701992207}+\frac{42387119144 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{4590701992207}+\frac{42387119144 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{4590701992207}+\frac{42387119144 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{4590701992207}+\frac{42387119144 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{4590701992207}-\frac{269010293109 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{4590701992207}-\frac{269010293109 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{4590701992207}-\frac{269010293109 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{4590701992207}-\frac{269010293109 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{4590701992207}-\frac{269010293109 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{4590701992207}-\frac{269010293109 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{4590701992207}-\frac{269010293109 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{4590701992207}-\frac{269010293109 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{4590701992207}+\frac{13102182087 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{4590701992207}+\frac{13102182087 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{4590701992207}+\frac{13102182087 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{4590701992207}+\frac{13102182087 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{4590701992207}+\frac{13102182087 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{4590701992207}+\frac{13102182087 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{4590701992207}+\frac{13102182087 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{4590701992207}+\frac{13102182087 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{4590701992207}+\frac{566392852412 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{4590701992207}+\frac{566392852412 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{4590701992207}+\frac{566392852412 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{4590701992207}+\frac{566392852412 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{4590701992207}+\frac{566392852412 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{4590701992207}+\frac{566392852412 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{4590701992207}+\frac{566392852412 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{4590701992207}+\frac{566392852412 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{4590701992207}+\frac{250319183702 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{4590701992207}+\frac{250319183702 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{4590701992207}+\frac{250319183702 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{4590701992207}+\frac{250319183702 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{4590701992207}+\frac{250319183702 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{4590701992207}+\frac{250319183702 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{4590701992207}+\frac{250319183702 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{4590701992207}+\frac{250319183702 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{4590701992207}+\frac{733632917937 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{4590701992207}+\frac{733632917937 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{4590701992207}+\frac{733632917937 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{4590701992207}+\frac{733632917937 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{4590701992207}+\frac{733632917937 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{4590701992207}+\frac{733632917937 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{4590701992207}+\frac{733632917937 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{4590701992207}+\frac{733632917937 \mathit{RootOf}\! \left(3 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}-5 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}-2 \textit{\_Z}^{2}-\textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{4590701992207}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 132
a(7) = 352
a(n+8) = -3*a(n)-4*a(n+1)+2*a(n+2)+7*a(n+3)+5*a(n+4)+4*a(n+5)+2*a(n+6)+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) = 51
a \! \left(6\right) = 132
a \! \left(7\right) = 352
a \! \left(n +8\right) = -3 a \! \left(n \right)-4 a \! \left(n +1\right)+2 a \! \left(n +2\right)+7 a \! \left(n +3\right)+5 a \! \left(n +4\right)+4 a \! \left(n +5\right)+2 a \! \left(n +6\right)+a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17077/
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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,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[64,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[15,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] = x^2
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[15,x]+F[34,x]
F[34,x] = F[18,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[2,x]*F[4,x]
F[41,x] = 2*F[22,x]+F[42,x]+F[59,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[39,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[22,x]+F[40,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[4,x]+F[51,x]
F[51,x] = F[22,x]+F[27,x]+F[52,x]
F[52,x] = F[15,x]*F[4,x]
F[53,x] = F[39,x]+F[54,x]
F[54,x] = 2*F[22,x]+F[42,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[22,x]+F[23,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[2,x]+F[47,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[56,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[39,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[69,x]
F[66,x] = F[2,x]+F[67,x]
F[67,x] = F[22,x]+F[48,x]+F[68,x]
F[68,x] = F[4,x]*F[58,x]
F[69,x] = F[56,x]+F[70,x]
F[70,x] = 2*F[22,x]+F[62,x]+F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[74,x]
F[73,x] = F[34,x]
F[74,x] = F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[39,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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \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_{64}\! \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_{15}\! \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) = x^{2}
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \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_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{59}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{22}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{53}\! \left(x \right) = F_{39}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{56}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{39}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{2}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{69}\! \left(x \right) = F_{56}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{62}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{34}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{39}\! \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_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(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_64(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_15(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), x**2)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_15(x) + F_34(x))
Eq(F_34(x), F_18(x))
Eq(F_35(x), F_36(x) + F_41(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_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_2(x)*F_4(x))
Eq(F_41(x), 2*F_22(x) + F_42(x) + F_59(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_39(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_22(x) + F_40(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_4(x) + F_51(x))
Eq(F_51(x), F_22(x) + F_27(x) + F_52(x))
Eq(F_52(x), F_15(x)*F_4(x))
Eq(F_53(x), F_39(x) + F_54(x))
Eq(F_54(x), 2*F_22(x) + F_42(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_22(x) + F_23(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_2(x) + F_47(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_56(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_39(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_69(x))
Eq(F_66(x), F_2(x) + F_67(x))
Eq(F_67(x), F_22(x) + F_48(x) + F_68(x))
Eq(F_68(x), F_4(x)*F_58(x))
Eq(F_69(x), F_56(x) + F_70(x))
Eq(F_70(x), 2*F_22(x) + F_62(x) + F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_74(x))
Eq(F_73(x), F_34(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_39(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, 2, 3], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 2, 3], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 2, 3], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [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, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "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, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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]]}], "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, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [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, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 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