0231_1032_1302_2013
Counting sequence:
1, 1, 2, 6, 20, 65, 206, 643, 1992, 6153, 18991, 58614, 180929, 558536, 1724282, 5323144, 16433406, 50732511, 156619245, 483508638, 1492669784, 4608117483, 14226023512, 43918104752, 135582517877, 418565867871, 1292182716611, 3989183770797, 12315276228131, 38019313433218, 117371966693683, 362346852350255, 1118625213766383, 3453382747447928, 10661168952581676, 32912808031924634, 101607331928846817, 313678793118545780, 968378791021253977, 2989546961648194366, 9229230461037930801, 28492174900012253715, 87960099594526524518, 271547509020822165875, 838312484812465087608, 2588010564805765215824, 7989620582884630012332, 24665292300785849070592, 76145874259220599984193, 235075023478084641402669, 725715834256515743348568, 2240405910837776547900556, 6916506996790492838602439, 21352411545264691871786952, 65918458408256999847826276, 203501283670450222062880723, 628242429442701357862700205, 1939489240722558804717435843, 5987527009621760439028467917, 18484495267214767049226752273, 57064722169039410422543584438, 176168322107522050089057780819, 543861014905946567207720877177, 1678989729799423398187663931889, 5183321538977104246612613744258, 16001778748005539519963778494898, 49400161879722581852958963103068, 152506545189356802565959660541793, 470813160131367986843964845055306, 1453478809566232413628094027107579, 4487131687798630465625447474842888, 13852524475162704445334430552194037, 42765055203700395499014013357055915, 132022863403319240033807890902592074, 407576615491035214042804288438359805, 1258257041340240381934488085760246276, 3884449504481246794904403523149486010, 11991943980533990243483675747323179156, 37021132664065930185763940471780708346, 114290415795399620457939663777372187627, 352833589971818390999028114349820106301, 1089256184309133078817365372827689366198, 3362715650600213061641220315179668608328, 10381264490101276265004439277972291599039, 32048696235794324565580031695713110798856, 98939674583341346690957996335977069491948, 305443289631368699469658303067645639005117, 942954417160985249446849018710192441930867, 2911057675932315848546505236763795412885527, 8986920935285988835040340282565912316568829, 27744124949779741113493474090635888107597207, 85650744539959120683146192414617004578816458, 264418144509098225130076309970838414950002155, 816302946590447954688161831713089199841198079, 2520063446664567383771352848200046829593993687, 7779856488011864296052287670688197885878552900, 24017715527825210481920791183317150969981373928, 74146696672929235989577606827799993175741241738, 228903228583005528497635019726232560711687648529, 706662473270418427402844212918001686050842833680, 2181585005244175584255017105794846791158051422321
Generating function in Maple syntax:
-(2*x-1)^2*(x-1)^4/(4*x^8-4*x^7-12*x^6+47*x^5-74*x^4+65*x^3-33*x^2+9*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}{4 x^{8}-4 x^{7}-12 x^{6}+47 x^{5}-74 x^{4}+65 x^{3}-33 x^{2}+9 x -1}
Generating function in sympy syntax:
-(x - 1)**4*(2*x - 1)**2/(4*x**8 - 4*x**7 - 12*x**6 + 47*x**5 - 74*x**4 + 65*x**3 - 33*x**2 + 9*x - 1)
Implicit equation for the generating function in Maple syntax:
(4*x^8-4*x^7-12*x^6+47*x^5-74*x^4+65*x^3-33*x^2+9*x-1)*F(x)+(2*x-1)^2*(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(4 x^{8}-4 x^{7}-12 x^{6}+47 x^{5}-74 x^{4}+65 x^{3}-33 x^{2}+9 x -1\right) F \! \left(x \right)+\left(2 x -1\right)^{2} \left(x -1\right)^{4} = 0
Explicit closed form in Maple syntax:
3752/3145*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+6)+3752/3145*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 8)^(-n+6)+3752/3145*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+6)+3752/3145*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+6)+3752/3145*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+6)+3752/3145*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+6)+3752/3145*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+6)+3752/3145*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+6)-22628/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+5)-22628/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+5)-22628/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+5)-22628/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+5)-22628/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+5)-22628/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+5)-22628/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+5)-22628/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 8)^(-n+5)-2524/629*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+4)-2524/629*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+4)-2524/629*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+4)-2524/629*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+4)-2524/629*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+4)-2524/629*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+4)-2524/629*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+4)-2524/629*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 8)^(-n+4)+294318/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+3)+294318/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+3)+294318/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+3)+294318/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+3)+294318/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+3)+294318/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+3)+294318/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+3)+294318/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 8)^(-n+3)-425521/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+2)-425521/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+2)-425521/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+2)-425521/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+2)-425521/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+2)-425521/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+2)-425521/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+2)-425521/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 8)^(-n+2)+63942/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+1)+63942/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+1)+63942/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+1)+63942/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+1)+63942/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+1)+63942/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+1)+63942/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+1)+63942/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 8)^(-n+1)+19622/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n-1)+19622/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n-1)+19622/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n-1)+19622/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n-1)+19622/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n-1)+19622/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n-1)+19622/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n-1)+19622/22015*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 8)^(-n-1)-24631/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n)-24631/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n)-24631/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n)-24631/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n)-24631/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n)-24631/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n)-24631/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n)-24631/4403*RootOf(4*_Z^8-4*_Z^7-12*_Z^6+47*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 8)^(-n)
Explicit closed form in latex syntax:
\frac{3752 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +6}}{3145}-\frac{22628 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +5}}{22015}-\frac{2524 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +4}}{629}+\frac{294318 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +3}}{22015}-\frac{425521 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +2}}{22015}+\frac{63942 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +1}}{4403}+\frac{19622 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =8\right)^{-n -1}}{22015}-\frac{24631 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf}\! \left(4 \textit{\_Z}^{8}-4 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+47 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =8\right)^{-n}}{4403}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 206
a(7) = 643
a(n+8) = 4*a(n)-4*a(n+1)-12*a(n+2)+47*a(n+3)-74*a(n+4)+65*a(n+5)-33*a(n+6)+9*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) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 206
a \! \left(7\right) = 643
a \! \left(n +8\right) = 4 a \! \left(n \right)-4 a \! \left(n +1\right)-12 a \! \left(n +2\right)+47 a \! \left(n +3\right)-74 a \! \left(n +4\right)+65 a \! \left(n +5\right)-33 a \! \left(n +6\right)+9 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15865/
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[19,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[6,x]
F[10,x] = F[11,x]+F[7,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[12,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[48,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[28,x]
F[24,x] = F[15,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[7,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[21,x]+F[22,x]+F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[2,x]+F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[2,x]+F[35,x]
F[39,x] = F[35,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[47,x]
F[43,x] = F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[2,x]+F[44,x]
F[47,x] = F[41,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[19,x]+F[50,x]
F[50,x] = F[35,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[29,x]
F[55,x] = F[52,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_{19}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{7}\! \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_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{48}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{19}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{29}\! \left(x \right)
F_{55}\! \left(x \right) = F_{52}\! \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_19(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_6(x))
Eq(F_10(x), F_11(x) + F_7(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_12(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_48(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_28(x))
Eq(F_24(x), F_15(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_7(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_21(x) + F_22(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_2(x) + F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_2(x) + F_35(x))
Eq(F_39(x), F_35(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_47(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_2(x) + F_44(x))
Eq(F_47(x), F_41(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_19(x) + F_50(x))
Eq(F_50(x), F_35(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_29(x))
Eq(F_55(x), F_52(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, 2, 3, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 2, 3, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 2, 3, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [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, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 0], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [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, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 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": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [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, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [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, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 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