0231_0321_1032_2103
Counting sequence:
1, 1, 2, 6, 20, 65, 202, 611, 1832, 5497, 16543, 49886, 150537, 454248, 1370410, 4133744, 12468490, 37608391, 113438977, 342172134, 1032116520, 3113241595, 9390667120, 28325638952, 85440317033, 257718730855, 777372459787, 2344835299765, 7072867893251, 21334317295698, 64351985477951, 194108766606671, 585501953463235, 1766084775359840, 5327147785821896, 16068596444044498, 48468674438437237, 146198979483238892, 440988779845720677, 1330180994645454822, 4012304982264863929, 12102556971941057987, 36505670906573655682, 110114252007331284619, 332144244826357563844, 1001866673570439220680, 3021990738197810080848, 9115412522113487378224, 27495367341148899473905, 82935931137587964278045, 250164640039734976954280, 754586658252967847752636, 2276105147086106209652119, 6865552927461254774030296, 20708980452909015736055888, 62465743972867749061493955, 188419182632226004131531317, 568340119333509933917296691, 1714318503729580939860108673, 5171002067698565565517920665, 15597604719292425446378110094, 47047993753282565608481009683, 141913694829761644779122674693, 428062817850326989345756265441, 1291191637606029234197293325790, 3894699038323554785671987189530, 11747815085948316975074913657340, 35435641608148693816405330191305, 106886658241930994353473787959886, 322408659520366791236887707791635, 972500641739979864514966948997176, 2933412209187044098573768287252309, 8848227774546118963323185815020699, 26689441908318334960368562569778666, 80504969755261383340583078657547877, 242831984930924774644460506264950140, 732468729381009152664618966557623130, 2209389507208632745942069783358168884, 6664314528060132676351433781273448222, 20101972958595851204079556982329028275, 60634790739047289320853219238165334201, 182896368209265994230471680183498988126, 551681321835547202416475209207158301976, 1664069570337139977360386217649708597151, 5019433186007869746491199560116874739376, 15140418380280022559064269251099453503452, 45668955086189235451489612130682682808057, 137754017509904201574715216018133719248523, 415515732827826037046304913326446454768367, 1253345109989492143684197055932546589386461, 3780540279531323453817572113838762673700647, 11403471151914897018521487820016913283986714, 34396976277865851585783794108697301674785359, 103753669501008781984143500096804277474819719, 312958437042956837114537911040937548912613043, 943995366982351334438126170666776695443318660, 2847430033534541149968876775841833764728513740, 8588874563858002824930837367514154695976223506, 25907139211465416983363288131248696479683313757, 78145263053040129866087734847458868160477099328, 235714259601703785319304550217117907276422498309
Generating function in Maple syntax:
-(2*x-1)*(x-1)^4/(2*x^6-7*x^5+20*x^4-27*x^3+19*x^2-7*x+1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x -1\right)^{4}}{2 x^{6}-7 x^{5}+20 x^{4}-27 x^{3}+19 x^{2}-7 x +1}
Generating function in sympy syntax:
(1 - 2*x)*(x - 1)**4/(2*x**6 - 7*x**5 + 20*x**4 - 27*x**3 + 19*x**2 - 7*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^6-7*x^5+20*x^4-27*x^3+19*x^2-7*x+1)*F(x)+(2*x-1)*(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{6}-7 x^{5}+20 x^{4}-27 x^{3}+19 x^{2}-7 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{4} = 0
Explicit closed form in Maple syntax:
-430200/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n+4)-430200/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n+4)-430200/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n+4)-430200/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n+4)-430200/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n+4)-430200/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n+4)+109450/152843*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n+3)+109450/152843*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n+3)+109450/152843*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n+3)+109450/152843*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n+3)+109450/152843*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n+3)+109450/152843*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n+3)-3494655/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n+2)-3494655/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n+2)-3494655/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n+2)-3494655/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n+2)-3494655/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n+2)-3494655/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n+2)+3393418/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n+1)+3393418/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n+1)+3393418/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n+1)+3393418/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n+1)+3393418/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n+1)+3393418/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n+1)+404170/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n-1)+404170/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n-1)+404170/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n-1)+404170/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n-1)+404170/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n-1)+404170/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n-1)-1552325/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n)-1552325/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n)-1552325/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n)-1552325/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n)-1552325/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n)-1552325/1681273*RootOf(2*_Z^6-7*_Z^5+20*_Z^4-27*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{430200 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{1681273}-\frac{430200 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{1681273}-\frac{430200 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{1681273}-\frac{430200 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{1681273}-\frac{430200 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{1681273}-\frac{430200 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{1681273}+\frac{109450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{152843}+\frac{109450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{152843}+\frac{109450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{152843}+\frac{109450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{152843}+\frac{109450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{152843}+\frac{109450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{152843}-\frac{3494655 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{1681273}-\frac{3494655 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{1681273}-\frac{3494655 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{1681273}-\frac{3494655 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{1681273}-\frac{3494655 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{1681273}-\frac{3494655 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{1681273}+\frac{3393418 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{1681273}+\frac{3393418 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{1681273}+\frac{3393418 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{1681273}+\frac{3393418 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{1681273}+\frac{3393418 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{1681273}+\frac{3393418 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{1681273}+\frac{404170 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{1681273}+\frac{404170 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{1681273}+\frac{404170 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{1681273}+\frac{404170 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{1681273}+\frac{404170 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{1681273}+\frac{404170 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{1681273}-\frac{1552325 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{1681273}-\frac{1552325 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{1681273}-\frac{1552325 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{1681273}-\frac{1552325 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{1681273}-\frac{1552325 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{1681273}-\frac{1552325 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-27 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{1681273}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+6) = -2*a(n)+7*a(n+1)-20*a(n+2)+27*a(n+3)-19*a(n+4)+7*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) = 20
a \! \left(5\right) = 65
a \! \left(n +6\right) = -2 a \! \left(n \right)+7 a \! \left(n +1\right)-20 a \! \left(n +2\right)+27 a \! \left(n +3\right)-19 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15802/
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[18,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[11,x]+F[17,x]
F[17,x] = F[15,x]
F[18,x] = F[19,x]+F[2,x]
F[19,x] = F[20,x]+F[21,x]+F[44,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[20,x]+F[25,x]+F[27,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[23,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[11,x]+F[17,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[2,x]+F[30,x]
F[33,x] = 2*F[20,x]+F[34,x]+F[36,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[29,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[20,x]+F[21,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[38,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[30,x]+F[41,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[18,x]+F[46,x]
F[46,x] = F[38,x]+F[47,x]
F[47,x] = 2*F[20,x]+F[42,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[54,x]
F[50,x] = F[17,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[50,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[30,x]+F[55,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[54,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_{18}\! \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_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{44}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \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_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \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_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{30}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{17}\! \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_{50}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{30}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{54}\! \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_18(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_11(x) + F_17(x))
Eq(F_17(x), F_15(x))
Eq(F_18(x), F_19(x) + F_2(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_44(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_20(x) + F_25(x) + F_27(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_23(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_11(x) + F_17(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_2(x) + F_30(x))
Eq(F_33(x), 2*F_20(x) + F_34(x) + F_36(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_29(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_20(x) + F_21(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_38(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_30(x) + F_41(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_18(x) + F_46(x))
Eq(F_46(x), F_38(x) + F_47(x))
Eq(F_47(x), 2*F_20(x) + F_42(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_54(x))
Eq(F_50(x), F_17(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_50(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_30(x) + F_55(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_54(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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"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": [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": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"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]]}], "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": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}], "requirements": [[{"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, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 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