0132_0231_1023_1302_2031
Counting sequence:
1, 1, 2, 6, 19, 57, 166, 482, 1407, 4124, 12108, 35567, 104500, 307089, 902566, 2653020, 7798858, 22926548, 67399512, 198144221, 582519255, 1712544458, 5034717751, 14801627670, 43515551986, 127932222147, 376110674011, 1105736240780, 3250779678504, 9557044234163, 28096983648936, 82603001493758, 242846570587813, 713950549469130, 2098960647491731, 6170785702728204, 18141643804173142, 53335062512988818, 156801055682594586, 460983262101510818, 1355255978490900201, 3984350234337416915, 11713688814555599230, 34437360575337876981, 101243239628154268150, 297647479370367839934, 875061113279790661783, 2572613595198073197425, 7563289706197320348582, 22235500616004631807430, 65370692761394570287035, 192184900440493511914340, 565009095012691936223956, 1661084074327172624637171, 4883461746621479281274092, 14357008774765832465246793, 42208521670397994428872666, 124089866458288071121109348, 364814837105248926381216870, 1072528073167743399246082892, 3153151546303915875887844048, 9270027445150533141936403661, 27253180689833185241212382119, 80122293284176349438613840714, 235553492055718131576094170139, 692509479513385900217448705114, 2035925576949059604902803001962, 5985467459287330701822125737055, 17596822355301650973827561554035, 51733328952207383444612096820812, 152092080629032561545891555758460, 447139232262399162653410920708599, 1314555578445041720912543657520088, 3864694135823192912723123611331070, 11361908928288503855785615149502729, 33403154287971819356192769202715346, 98202751265506213034059822638951139, 288708673228130511777665568651981320, 848781698302837070165219361725671638, 2495354099752250715961003542474844166, 7336152623932646003217953665801922490, 21567734746334092687791250829669958770, 63407511530051515548405058659513952201, 186413295866270321268528894568203290043, 548041013394070396018748713456889836870, 1611199195670381786080484802489206577033, 4736803970293827259259691792124016192650, 13925845986818365844617859861586179116310, 40940935631869862404752863172862382012907, 120363259223137738181209352834919628021145, 353858893237867713990878422370864557756998, 1040318425503868755463679772131091764931578, 3058457614389643739731743922890067892225239, 8991634435857835612849834573740538675349964, 26434726264545293809217034186171884639544460, 77716099077016485199616130949723180313199687, 228479462783365939612282559101740347151129028, 671712367627234813440492406295028846542513057, 1974783638436645177943906816186303523134643454, 5805714777014858988263555281317067789684860332, 17068363042916742763905123329235169122487699458
Generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x-1)^3/(3*x^6-12*x^5+25*x^4-28*x^3+19*x^2-7*x+1)
Generating function in latex syntax:
\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}{3 x^{6}-12 x^{5}+25 x^{4}-28 x^{3}+19 x^{2}-7 x +1}
Generating function in sympy syntax:
(x - 1)**3*(x**3 - 2*x**2 + 3*x - 1)/(3*x**6 - 12*x**5 + 25*x**4 - 28*x**3 + 19*x**2 - 7*x + 1)
Implicit equation for the generating function in Maple syntax:
(3*x^6-12*x^5+25*x^4-28*x^3+19*x^2-7*x+1)*F(x)-(x^3-2*x^2+3*x-1)*(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x^{6}-12 x^{5}+25 x^{4}-28 x^{3}+19 x^{2}-7 x +1\right) F \! \left(x \right)-\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1/3,0)-21699/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n+4)-21699/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n+4)-21699/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n+4)-21699/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n+4)-21699/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n+4)-21699/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n+4)+77949/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n+3)+77949/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n+3)+77949/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n+3)+77949/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n+3)+77949/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n+3)+77949/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n+3)-141553/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n+2)-141553/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n+2)-141553/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n+2)-141553/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n+2)-141553/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n+2)-141553/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n+2)+123205/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n+1)+123205/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n+1)+123205/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n+1)+123205/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n+1)+123205/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n+1)+123205/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n+1)+13060/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n-1)+13060/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n-1)+13060/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n-1)+13060/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n-1)+13060/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n-1)+13060/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n-1)-57183/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 1)^(-n)-57183/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 2)^(-n)-57183/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 3)^(-n)-57183/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 4)^(-n)-57183/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 5)^(-n)-57183/38023*RootOf(3*_Z^6-12*_Z^5+25*_Z^4-28*_Z^3+19*_Z^2-7*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{21699 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{38023}+\frac{77949 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{38023}-\frac{141553 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{38023}+\frac{123205 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{38023}+\frac{13060 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{38023}-\frac{57183 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-12 \textit{\_Z}^{5}+25 \textit{\_Z}^{4}-28 \textit{\_Z}^{3}+19 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{38023}+\left(\left\{\begin{array}{cc}\frac{1}{3} & 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) = 19
a(5) = 57
a(6) = 166
a(n+6) = -3*a(n)+12*a(n+1)-25*a(n+2)+28*a(n+3)-19*a(n+4)+7*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) = 19
a \! \left(5\right) = 57
a \! \left(6\right) = 166
a \! \left(n +6\right) = -3 a \! \left(n \right)+12 a \! \left(n +1\right)-25 a \! \left(n +2\right)+28 a \! \left(n +3\right)-19 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18780/
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[22,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[57,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[32,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[1,x]+F[28,x]
F[31,x] = F[15,x]
F[32,x] = F[33,x]+F[37,x]
F[33,x] = F[24,x]+F[25,x]+F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[2,x]+F[36,x]
F[36,x] = F[34,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[50,x]
F[40,x] = F[41,x]+F[48,x]
F[41,x] = F[24,x]+F[42,x]+F[47,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[21,x]+F[4,x]
F[45,x] = F[41,x]+F[46,x]
F[46,x] = F[38,x]
F[47,x] = F[2,x]*F[4,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[40,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[41,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[53,x]
F[56,x] = F[51,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[2,x]+F[60,x]
F[60,x] = F[24,x]+F[42,x]+F[61,x]
F[61,x] = F[4,x]*F[59,x]
F[62,x] = F[63,x]+F[78,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[69,x]
F[66,x] = F[2,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[66,x]
F[69,x] = F[63,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[77,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[47,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[73,x]
F[77,x] = F[71,x]
F[78,x] = F[51,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_{22}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \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_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{57}\! \left(x \right)
F_{24}\! \left(x \right) = 0
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_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{50}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{41}\! \left(x \right) = F_{24}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{38}\! \left(x \right)
F_{47}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{51}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = F_{2}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{24}\! \left(x \right)+F_{42}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{78}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{2}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{63}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{77}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{47}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{77}\! \left(x \right) = F_{71}\! \left(x \right)
F_{78}\! \left(x \right) = F_{51}\! \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_22(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_57(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_32(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_1(x) + F_28(x))
Eq(F_31(x), F_15(x))
Eq(F_32(x), F_33(x) + F_37(x))
Eq(F_33(x), F_24(x) + F_25(x) + F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_2(x) + F_36(x))
Eq(F_36(x), F_34(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_50(x))
Eq(F_40(x), F_41(x) + F_48(x))
Eq(F_41(x), F_24(x) + F_42(x) + F_47(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_21(x) + F_4(x))
Eq(F_45(x), F_41(x) + F_46(x))
Eq(F_46(x), F_38(x))
Eq(F_47(x), F_2(x)*F_4(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_40(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_41(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_53(x))
Eq(F_56(x), F_51(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), F_2(x) + F_60(x))
Eq(F_60(x), F_24(x) + F_42(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_59(x))
Eq(F_62(x), F_63(x) + F_78(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_69(x))
Eq(F_66(x), F_2(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_66(x))
Eq(F_69(x), F_63(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_77(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_47(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_73(x))
Eq(F_77(x), F_71(x))
Eq(F_78(x), F_51(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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]]}], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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]]}], "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, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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]]}], "requirements": []}], "class_module": 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