0231_0312_1302_2031_2103
Counting sequence:
1, 1, 2, 6, 19, 57, 165, 471, 1342, 3830, 10947, 31309, 89555, 256143, 732566, 2095068, 5991657, 17135501, 49005847, 140152093, 400821870, 1146312940, 3278347053, 9375763241, 26813797789, 76684930645, 219311664274, 627210663418, 1793763309031, 5129993790345, 14671298130969, 41958528141899, 119997430905550, 343181328364418, 981466213487119, 2806900750729237, 8027471263087695, 22957810269194259, 65657170867702454, 187773312689758208, 537013954340617501, 1535809232023388781, 4392269471031704763, 12561476193720104385, 35924636501943490558, 102741070228829824792, 293829765297535582105, 840325400373125248241, 2403251344523249081049, 6873072053264470612137, 19656337468412511833890, 56215270213645513252558, 160770367840474482651835, 459788080308603185637529, 1314950520009002494148173, 3760634396853877360515391, 10755059488248807714959102, 30758455193767460597985358, 87966279214048897986866523, 251575257275338194149469053, 719481494939082075684860075, 2057649178882591690574489367, 5884682473612037040035647574, 16829636543797159208942947204, 48131172321768656644210181777, 137650610756749470097929436685, 393667756834099274464713152351, 1125852634570984693749534618213, 3219832289451634496221242225198, 9208416495952781432905671217284, 26335202190725473458785939552325, 75316193042659709345993398365401, 215397204599283080409866425242933, 616015678366886738646211787430269, 1761746707436511833797170634712882, 5038429329252976441614025649628034, 14409432410868533737164360702530447, 41209616893480000267304269186689513, 117855615411088260179342160893722049, 337055938176507271564581205670294067, 963948175602644927707973902226031182, 2756800815540215142684660152151829338, 7884190176315057702876351446232273575, 22548039882280021774302785722923000581, 64485266230667579262827964153850863207, 184421776019122013407909272020327538459, 527428875743281656713843271823680879862, 1508396811767925158202743180984339417448, 4313872535979792755248140013019052738885, 12337268357700492104989608413640978242029, 35283423249163609126443424560714396393859, 100907260836438199928212821378057383461257, 288585243489783678847066646227685405470846, 825326562922460371544285322886749079318320, 2360356084837428961279123715396073980931633, 6750395658539466292858754202512416853448801, 19305494556329617998818146156709150873334321, 55211892593731512364091899268895288769424721, 157900802535115209780980556211724907257072194, 451581394332861146059682740272731341340055830, 1291480172573919879951367901435611017658079075
Generating function in Maple syntax:
-(x^2+x-1)*(x-1)^3/(x^6-3*x^5+2*x^4+5*x^3-8*x^2+5*x-1)
Generating function in latex syntax:
-\frac{\left(x^{2}+x -1\right) \left(x -1\right)^{3}}{x^{6}-3 x^{5}+2 x^{4}+5 x^{3}-8 x^{2}+5 x -1}
Generating function in sympy syntax:
(x - 1)**3*(-x**2 - x + 1)/(x**6 - 3*x**5 + 2*x**4 + 5*x**3 - 8*x**2 + 5*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^6-3*x^5+2*x^4+5*x^3-8*x^2+5*x-1)*F(x)+(x^2+x-1)*(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}-3 x^{5}+2 x^{4}+5 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x^{2}+x -1\right) \left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
9515/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n)+9515/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n)+9515/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n)+9515/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n)+9515/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n)+9515/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n)+1133/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n-1)+1133/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n-1)+1133/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n-1)+1133/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n-1)+1133/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n-1)+1133/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n-1)-9277/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+1)-9277/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+1)-9277/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+1)-9277/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+1)-9277/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+1)-9277/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+1)+45/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+2)+45/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+2)+45/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+2)+45/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+2)+45/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+2)+45/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+2)+3819/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+3)+3819/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+3)+3819/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+3)+3819/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+3)+3819/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+3)+3819/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+3)-1821/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+4)-1821/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+4)-1821/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+4)-1821/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+4)-1821/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+4)-1821/32266*RootOf(_Z^6-3*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+4)
Explicit closed form in latex syntax:
\frac{9515 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{32266}+\frac{9515 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{32266}+\frac{9515 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{32266}+\frac{9515 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{32266}+\frac{9515 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{32266}+\frac{9515 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{32266}+\frac{1133 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{32266}+\frac{1133 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{32266}+\frac{1133 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{32266}+\frac{1133 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{32266}+\frac{1133 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{32266}+\frac{1133 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{32266}-\frac{9277 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{32266}-\frac{9277 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{32266}-\frac{9277 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{32266}-\frac{9277 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{32266}-\frac{9277 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{32266}-\frac{9277 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{32266}+\frac{45 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{32266}+\frac{45 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{32266}+\frac{45 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{32266}+\frac{45 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{32266}+\frac{45 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{32266}+\frac{45 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{32266}+\frac{3819 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{32266}+\frac{3819 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{32266}+\frac{3819 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{32266}+\frac{3819 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{32266}+\frac{3819 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{32266}+\frac{3819 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{32266}-\frac{1821 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{32266}-\frac{1821 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{32266}-\frac{1821 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{32266}-\frac{1821 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{32266}-\frac{1821 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{32266}-\frac{1821 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{32266}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(n+6) = a(n)-3*a(n+1)+2*a(n+2)+5*a(n+3)-8*a(n+4)+5*a(n+5), n >= 6
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) = 57
a \! \left(n +6\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+2 a \! \left(n +2\right)+5 a \! \left(n +3\right)-8 a \! \left(n +4\right)+5 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20738/
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[17,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[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[41,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[26,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]+F[38,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[2,x]+F[27,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[2,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[27,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[26,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[17,x]+F[43,x]
F[43,x] = F[32,x]+F[44,x]
F[44,x] = F[36,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_{17}\! \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_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{38}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \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_{2}\! \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_{27}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{26}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{32}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \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_17(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))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_41(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_26(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x) + F_38(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_2(x) + F_27(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_2(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_27(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_26(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_17(x) + F_43(x))
Eq(F_43(x), F_32(x) + F_44(x))
Eq(F_44(x), F_36(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": [0, 3, 1, 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, 3, 1], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": 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"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": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": 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{"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, "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": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], 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