0321_1032_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 65, 203, 618, 1862, 5603, 16894, 51041, 154370, 467007, 1412689, 4272737, 12921869, 39077817, 118177690, 357392160, 1080831466, 3268680476, 9885239729, 29895216466, 90409888568, 273419837892, 826883011665, 2500680037229, 7562618701628, 22871060060344, 69167230919243, 209177266921944, 632599112186405, 1913122021615830, 5785711353301377, 17497292633979507, 52915748967105943, 160029128382957077, 483964083162930871, 1463616256402610054, 4426304803437759680, 13386141433570077888, 40482702939644176506, 122428800369881136603, 370252232968516406046, 1119726041618658786158, 3386303434897989745560, 10240943344167962924705, 30970916397403640368419, 93663018167283546121774, 283258036657250151868306, 856636022422598668107319, 2590660034122708501571062, 7834738718341837546546573, 23694012327430487548991810, 71656023302746063584418511, 216703933660885222792814637, 655361443457352531911130261, 1981960430135138418500686937, 5993894187455485241768318281, 18126884363661593866165755270, 54819776001592865401288890800, 165787334468203734721069844218, 501378193688230805859790881368, 1516280443933943235059748678477, 4585573153359271203884390324718, 13867804751378401672087228831464, 41939361163929460927300524662176, 126834062519064989155455809771145, 383574736682585280367013905206117, 1160016289780191625586385116320024, 3508150208726963228231842041403332, 10609435397948660198845356222370639, 32085319261198863276237921175035592, 97033223114974325792694470018556265, 293450294554699750227255906779609230, 887459703077212686726772319541957921, 2683877778282778519929258187365519351, 8116650146235873529602777833770124859, 24546575902030357937604856200046660721, 74234367338546066021548557245592906383, 224501426029785958259519647684172396362, 678942814445417677298663312605447488840, 2053275800688705074705441988963047461424, 6209567910572202640881920281622907454798, 18779130218685063221335931674669142685559, 56792314191444892056005323904509610360458, 171752733681487665283708679426943388320694, 519418902839985208472191244897959851901580, 1570839606709409264280728716673257210822545, 4750572334806100456236221189129629305821387, 14366799393033093679718768503796567239832882, 43448433210328266760541021323059289227544878, 131397835856731783283791608678912793638605387, 397376613887389086788029415708304610728379726, 1201756271212720635167890503570106828988813769, 3634381302087327043865862197280606947682792530, 10991186620255982467569749427787417091469245103, 33239820833304350753309204182684324635674927721, 100524695576904087111605417100807573478690667833, 304009292694634744208545786733427254465342181701
Generating function in Maple syntax:
(2*x-1)*(x-1)^5/(8*x^6-27*x^5+47*x^4-46*x^3+26*x^2-8*x+1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x -1\right)^{5}}{8 x^{6}-27 x^{5}+47 x^{4}-46 x^{3}+26 x^{2}-8 x +1}
Generating function in sympy syntax:
(x - 1)**5*(2*x - 1)/(8*x**6 - 27*x**5 + 47*x**4 - 46*x**3 + 26*x**2 - 8*x + 1)
Implicit equation for the generating function in Maple syntax:
(8*x^6-27*x^5+47*x^4-46*x^3+26*x^2-8*x+1)*F(x)-(2*x-1)*(x-1)^5 = 0
Implicit equation for the generating function in latex syntax:
\left(8 x^{6}-27 x^{5}+47 x^{4}-46 x^{3}+26 x^{2}-8 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{5} = 0
Explicit closed form in Maple syntax:
-733536/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+4)-733536/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+4)-733536/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+4)-733536/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+4)-733536/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+4)-733536/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+4)+2004596/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+3)+2004596/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+3)+2004596/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+3)+2004596/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+3)+2004596/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+3)+2004596/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+3)-3038506/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+2)-3038506/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+2)-3038506/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+2)-3038506/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+2)-3038506/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+2)-3038506/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+2)+2306181/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+1)+2306181/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+1)+2306181/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+1)+2306181/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+1)+2306181/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+1)+2306181/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+1)+181346/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n-1)+181346/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n-1)+181346/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n-1)+181346/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n-1)+181346/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n-1)+181346/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n-1)-918729/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n)-918729/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n)-918729/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n)-918729/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n)-918729/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n)-918729/468893*RootOf(8*_Z^6-27*_Z^5+47*_Z^4-46*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n)+piecewise(n = 0,1/4,0)
Explicit closed form in latex syntax:
-\frac{733536 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{468893}-\frac{733536 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{468893}-\frac{733536 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{468893}-\frac{733536 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{468893}-\frac{733536 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{468893}-\frac{733536 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{468893}+\frac{2004596 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{468893}+\frac{2004596 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{468893}+\frac{2004596 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{468893}+\frac{2004596 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{468893}+\frac{2004596 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{468893}+\frac{2004596 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{468893}-\frac{3038506 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{468893}-\frac{3038506 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{468893}-\frac{3038506 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{468893}-\frac{3038506 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{468893}-\frac{3038506 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{468893}-\frac{3038506 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{468893}+\frac{2306181 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{468893}+\frac{2306181 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{468893}+\frac{2306181 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{468893}+\frac{2306181 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{468893}+\frac{2306181 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{468893}+\frac{2306181 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{468893}+\frac{181346 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{468893}+\frac{181346 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{468893}+\frac{181346 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{468893}+\frac{181346 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{468893}+\frac{181346 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{468893}+\frac{181346 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{468893}-\frac{918729 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{468893}-\frac{918729 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{468893}-\frac{918729 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{468893}-\frac{918729 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{468893}-\frac{918729 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{468893}-\frac{918729 \mathit{RootOf}\! \left(8 \textit{\_Z}^{6}-27 \textit{\_Z}^{5}+47 \textit{\_Z}^{4}-46 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{468893}+\left(\left\{\begin{array}{cc}\frac{1}{4} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 203
a(n+6) = -8*a(n)+27*a(n+1)-47*a(n+2)+46*a(n+3)-26*a(n+4)+8*a(n+5), n >= 7
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) = 203
a \! \left(n +6\right) = -8 a \! \left(n \right)+27 a \! \left(n +1\right)-47 a \! \left(n +2\right)+46 a \! \left(n +3\right)-26 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15938/
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[23,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[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]+F[16,x]+F[21,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[15,x]+F[16,x]+F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]+F[18,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[10,x]
F[23,x] = F[2,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[57,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[38,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[15,x]+F[19,x]+F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[27,x]+F[31,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[11,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[31,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[2,x]+F[39,x]
F[42,x] = 2*F[15,x]+F[43,x]+F[52,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[38,x]+F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[39,x]+F[46,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[45,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[39,x]+F[54,x]
F[54,x] = 2*F[15,x]+F[52,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[39,x]+F[54,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[23,x]+F[59,x]
F[59,x] = F[39,x]+F[60,x]
F[60,x] = 2*F[15,x]+F[55,x]+F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[59,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_{23}\! \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_{14}\! \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_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{57}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{38}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \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_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \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_{11}\! \left(x \right)+F_{32}\! \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_{31}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{52}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \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_{45}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \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_{15}\! \left(x \right)+F_{52}\! \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_{39}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{23}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{39}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{59}\! \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_23(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_14(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_11(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_21(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x) + F_18(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_10(x))
Eq(F_23(x), F_2(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_57(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_38(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_15(x) + F_19(x) + F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_27(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_11(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_31(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_2(x) + F_39(x))
Eq(F_42(x), 2*F_15(x) + F_43(x) + F_52(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_38(x) + F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_39(x) + F_46(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_45(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_39(x) + F_54(x))
Eq(F_54(x), 2*F_15(x) + F_52(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_39(x) + F_54(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_23(x) + F_59(x))
Eq(F_59(x), F_39(x) + F_60(x))
Eq(F_60(x), 2*F_15(x) + F_55(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_59(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, 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": [1, 3, 0, 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, 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": [1, 3, 0, 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, 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": [1, 3, 0, 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, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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": [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": [1, 3, 0, 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, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [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": [1, 3, 0, 2], "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]]}], "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": [2, 1, 0], "pos": [[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]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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], 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