0123_0231_1302
Counting sequence:
1, 1, 2, 6, 21, 75, 264, 918, 3176, 10978, 37964, 131362, 454692, 1574092, 5449596, 18867020, 65319484, 226142468, 782927696, 2710574976, 9384289408, 32489384592, 112481640304, 389423203608, 1348223897824, 4667692322496, 16160039700672, 55947750119968, 193696971369872, 670599204845072, 2321684694012992, 8037915614140224, 27828105854674496, 96343817560295072, 333552388752567936, 1154793310676257120, 3998015410341539456, 13841548157224792960, 47920889672768352576, 165907139934517099840, 574387897829483214016, 1988591071506884686144, 6884710601704146925568, 23835589301576093785984, 82521307026731177076224, 285697409333605481795072, 989114358955728747344128, 3424417524031517864291200, 11855692188388330008567552, 41045648283079652322868480, 142104334036974129029972480, 491980090381868833109065728, 1703286609605969532684783872, 5896956667923562911256478464, 20415881712011256747915701248, 70681921124850701948422493696, 244708215122561311600493108224, 847205474829915085112218572288, 2933114101716182973734679335936, 10154748274512154751902755080704, 35156802273178557490804191689728, 121716532272657622264181833745408, 421395384977414074366003430415360, 1458914965491119396770459548218368, 5050916437179381362719180895435776, 17486801807383439898445686563062784, 60541139663255289791453808652349440, 209599767418774972216383307227525120, 725656351141810346743055289983225856, 2512298302795149130920751723981946880, 8697839896661971247420797754023030784, 30112832852609450116077245207040317440, 104253781764504209137441254958560116736, 360937513431554213078041611858225119232, 1249603481017405565192462672861017735168, 4326258151792059747576791485094040748032, 14977958912781348789622301476706678476800, 51855262751724261947021165321534305521664, 179528351673855053425115159363256262033408, 621545959742719971392434542272927775080448, 2151857221829319939265433013197938285281280, 7449955116844012157893657630904383262187520, 25792525024410076116894254840400701514973184, 89296423495318777770630786135973708404449280, 309153572265951364351397838365191121553850368, 1070322052145898407553084061294696021224456192, 3705567064657129866161199245239463068695478272, 12829061349471213223941391893324814829416267776, 44415554282708661099206759344914046028225822720, 153771301617600567277350205243265957997298204672, 532372354303287792994043991897395505205615001600, 1843128858538486060443537197074397130146743091200, 6381105182712157235405151418600435337508368744448, 22092054586527281281160281390175233770646331785216, 76485007201630504922267812930285476092411866054656, 264799107014747016207847899138098173886036987445248, 916762247154664020077228857090379371454500934582272, 3173926933829362673191799103765904951234639867215872, 10988467525308051847680718377946227091310342386614272, 38043225654557949380463423779097510697787206273204224, 131709632382341023524346452264633002021245988100702208
Generating function in Maple syntax:
-(3*x-1)*(x-1)^4/(2*x^6-10*x^5+28*x^4-36*x^3+24*x^2-8*x+1)
Generating function in latex syntax:
-\frac{\left(3 x -1\right) \left(x -1\right)^{4}}{2 x^{6}-10 x^{5}+28 x^{4}-36 x^{3}+24 x^{2}-8 x +1}
Generating function in sympy syntax:
(1 - 3*x)*(x - 1)**4/(2*x**6 - 10*x**5 + 28*x**4 - 36*x**3 + 24*x**2 - 8*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^6-10*x^5+28*x^4-36*x^3+24*x^2-8*x+1)*F(x)+(3*x-1)*(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{6}-10 x^{5}+28 x^{4}-36 x^{3}+24 x^{2}-8 x +1\right) F \! \left(x \right)+\left(3 x -1\right) \left(x -1\right)^{4} = 0
Explicit closed form in Maple syntax:
-2228/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 1)^(-n+4)-2228/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 2)^(-n+4)-2228/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 3)^(-n+4)-2228/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 4)^(-n+4)-2228/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 5)^(-n+4)-2228/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 6)^(-n+4)+9450/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 1)^(-n+3)+9450/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 2)^(-n+3)+9450/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 3)^(-n+3)+9450/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 4)^(-n+3)+9450/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 5)^(-n+3)+9450/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 6)^(-n+3)-24203/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 1)^(-n+2)-24203/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 2)^(-n+2)-24203/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 3)^(-n+2)-24203/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 4)^(-n+2)-24203/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 5)^(-n+2)-24203/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 6)^(-n+2)+21667/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 1)^(-n+1)+21667/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 2)^(-n+1)+21667/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 3)^(-n+1)+21667/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 4)^(-n+1)+21667/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 5)^(-n+1)+21667/12469*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 6)^(-n+1)+2893/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 1)^(-n-1)+2893/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 2)^(-n-1)+2893/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 3)^(-n-1)+2893/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 4)^(-n-1)+2893/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 5)^(-n-1)+2893/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 6)^(-n-1)-15095/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 1)^(-n)-15095/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 2)^(-n)-15095/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 3)^(-n)-15095/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 4)^(-n)-15095/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 5)^(-n)-15095/24938*RootOf(2*_Z^6-10*_Z^5+28*_Z^4-36*_Z^3+24*_Z^2-8*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{2228 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{12469}+\frac{9450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{12469}-\frac{24203 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{12469}+\frac{21667 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{12469}+\frac{2893 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{24938}-\frac{15095 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-10 \textit{\_Z}^{5}+28 \textit{\_Z}^{4}-36 \textit{\_Z}^{3}+24 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{24938}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(n+6) = -2*a(n)+10*a(n+1)-28*a(n+2)+36*a(n+3)-24*a(n+4)+8*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) = 21
a \! \left(5\right) = 75
a \! \left(n +6\right) = -2 a \! \left(n \right)+10 a \! \left(n +1\right)-28 a \! \left(n +2\right)+36 a \! \left(n +3\right)-24 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14639/
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[12,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[14,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[10,x]+F[20,x]
F[20,x] = F[18,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[15,x]+F[23,x]+F[64,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[34,x]
F[25,x] = F[26,x]+F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[10,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[10,x]+F[31,x]
F[34,x] = F[22,x]+F[35,x]
F[35,x] = 2*F[15,x]+F[36,x]+F[40,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[26,x]
F[39,x] = F[35,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[56,x]
F[42,x] = F[43,x]
F[43,x] = F[15,x]+F[44,x]+F[54,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[48,x]
F[46,x] = F[10,x]+F[47,x]
F[47,x] = F[27,x]
F[48,x] = F[43,x]+F[49,x]
F[49,x] = 2*F[15,x]+F[40,x]+F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[47,x]
F[53,x] = F[49,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[2,x]+F[43,x]
F[56,x] = F[57,x]
F[57,x] = 2*F[15,x]+F[58,x]+F[62,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[31,x]
F[61,x] = F[57,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[43,x]+F[57,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[55,x]+F[66,x]
F[66,x] = F[43,x]+F[67,x]
F[67,x] = F[15,x]+F[58,x]+F[68,x]+F[70,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[43,x]+F[67,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[43,x]+F[72,x]
F[72,x] = F[70,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{64}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{22}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{56}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{54}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{27}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{47}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right)+F_{43}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{31}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{43}\! \left(x \right)+F_{57}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{55}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{43}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{43}\! \left(x \right)+F_{67}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{43}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{70}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_21(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_14(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_10(x) + F_20(x))
Eq(F_20(x), F_18(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_64(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_34(x))
Eq(F_25(x), F_26(x) + F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_10(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_31(x))
Eq(F_34(x), F_22(x) + F_35(x))
Eq(F_35(x), 2*F_15(x) + F_36(x) + F_40(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_26(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_56(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_15(x) + F_44(x) + F_54(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_48(x))
Eq(F_46(x), F_10(x) + F_47(x))
Eq(F_47(x), F_27(x))
Eq(F_48(x), F_43(x) + F_49(x))
Eq(F_49(x), 2*F_15(x) + F_40(x) + F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_47(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_2(x) + F_43(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), 2*F_15(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_31(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_43(x) + F_57(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_55(x) + F_66(x))
Eq(F_66(x), F_43(x) + F_67(x))
Eq(F_67(x), F_15(x) + F_58(x) + F_68(x) + F_70(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_43(x) + F_67(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_43(x) + F_72(x))
Eq(F_72(x), F_70(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, 2, 3], "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, 3, 0, 2], "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, 2, 3], "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, 3, 0, 2], "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, 2, 3], "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, 3, 0, 2], "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, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, 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