0123_0132_1203_2013_2103
Counting sequence:
1, 1, 2, 6, 19, 60, 191, 610, 1950, 6236, 19946, 63802, 204091, 652857, 2088402, 6680529, 21370166, 68360472, 218676571, 699518937, 2237673424, 7158036932, 22897663397, 73246756698, 234307198774, 749519376439, 2397618590612, 7669681514455, 24534350360862, 78482313313478, 251055088577808, 803093777945874, 2568996389713322, 8217897624909227, 26288025021719761, 84092098865761794, 269000089805395621, 860497588849848458, 2752623989650249936, 8805299313534964135, 28167049438085712129, 90102862582766002112, 288227769949927876120, 922004528923794389737, 2949376985790333730738, 9434687500356791187222, 30180383402407305947691, 96543265718324704106732, 308829812772212900420079, 987907883033388569954090, 3160193559678652776334094, 10109063310609201383553572, 32337627138350561511573418, 103444018185297969703464666, 330904455436369522252636875, 1058521899560188238188092297, 3386078952520985496738752274, 10831642384979940494887254713, 34649067077642751913724103038, 110838024990169594507286548520, 354556956936031832946738670307, 1134183288838723112631175198457, 3628110258496792048612578694864, 11605870212818325030624716176700, 37125724909085689700718214013965, 118760543134696991534485068734090, 379900100003072810100126595064422, 1215252828699628139533134372809071, 3887441560690566717487513060401476, 12435438561336247612960157203606599, 39779410133511611453297916534545862, 127249349732631354942519765429954038, 407054728891931647039228278302520984, 1302117084774336791646815968233085170, 4165309434131044125676374605930524234, 13324303079141215633328824499462097435, 42622776375280121763206825822014799649, 136344922143142075898987877738772244866, 436150325603872611852808475005771270221, 1395190253764299839514427264077605629586, 4463038842178518289714135766167689040928, 14276702158040727021800128453220864321039, 45669381718827517245994913119513408904001, 146090630979879491941636981191777959358976, 467325627298797965910881786393183237328752, 1494916138464029872189939256215252094670673, 4782049454376145058409400423007789876945762, 15297177143057130810079439626971040165301990, 48933753357972557089785805069238291333243523, 156532946916005237039103636512062013447004060, 500729287838632975321577753037577274616748319, 1601770263955516450584589836641649884952780850, 5123862415890774249127421568077862185267810558, 16390593987020821241859892581573602777583411308, 52431456866247356017574372975592576375940547882, 167721662271303068755146982855839165687866211706, 536520586616734573659142299412555609839213441307, 1716262145064707261058154770375987812617267899737, 5490107600822181872048036284882516250822760863698, 17562166452997828137659814037923260312492487348641, 56179170418556609070522183161727359451030909614406
Generating function in Maple syntax:
(x^6+2*x^5+2*x^4-x^3-2*x^2-2*x+1)/(x^6+2*x^5+3*x^4-x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{6}+2 x^{5}+2 x^{4}-x^{3}-2 x^{2}-2 x +1}{x^{6}+2 x^{5}+3 x^{4}-x^{2}-3 x +1}
Generating function in sympy syntax:
(x**6 + 2*x**5 + 2*x**4 - x**3 - 2*x**2 - 2*x + 1)/(x**6 + 2*x**5 + 3*x**4 - x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^6+2*x^5+3*x^4-x^2-3*x+1)*F(x)-x^6-2*x^5-2*x^4+x^3+2*x^2+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}+2 x^{5}+3 x^{4}-x^{2}-3 x +1\right) F \! \left(x \right)-x^{6}-2 x^{5}-2 x^{4}+x^{3}+2 x^{2}+2 x -1 = 0
Explicit closed form in Maple syntax:
-7400/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 1)^(-n+4)-7400/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 2)^(-n+4)-7400/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 3)^(-n+4)-7400/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 4)^(-n+4)-7400/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 5)^(-n+4)-7400/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 6)^(-n+4)-2511/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 1)^(-n+3)-2511/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 2)^(-n+3)-2511/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 3)^(-n+3)-2511/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 4)^(-n+3)-2511/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 5)^(-n+3)-2511/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 6)^(-n+3)-1365/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 1)^(-n+2)-1365/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 2)^(-n+2)-1365/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 3)^(-n+2)-1365/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 4)^(-n+2)-1365/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 5)^(-n+2)-1365/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 6)^(-n+2)+30649/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 1)^(-n+1)+30649/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 2)^(-n+1)+30649/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 3)^(-n+1)+30649/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 4)^(-n+1)+30649/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 5)^(-n+1)+30649/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 6)^(-n+1)+10752/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 1)^(-n-1)+10752/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 2)^(-n-1)+10752/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 3)^(-n-1)+10752/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 4)^(-n-1)+10752/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 5)^(-n-1)+10752/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 6)^(-n-1)-3763/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 1)^(-n)-3763/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 2)^(-n)-3763/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 3)^(-n)-3763/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 4)^(-n)-3763/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 5)^(-n)-3763/224629*RootOf(_Z^6+2*_Z^5+3*_Z^4-_Z^2-3*_Z+1,index = 6)^(-n)+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
-\frac{7400 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{224629}-\frac{7400 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{224629}-\frac{7400 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{224629}-\frac{7400 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{224629}-\frac{7400 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{224629}-\frac{7400 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{224629}-\frac{2511 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{224629}-\frac{2511 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{224629}-\frac{2511 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{224629}-\frac{2511 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{224629}-\frac{2511 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{224629}-\frac{2511 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{224629}-\frac{1365 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{224629}-\frac{1365 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{224629}-\frac{1365 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{224629}-\frac{1365 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{224629}-\frac{1365 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{224629}-\frac{1365 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{224629}+\frac{30649 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{224629}+\frac{30649 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{224629}+\frac{30649 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{224629}+\frac{30649 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{224629}+\frac{30649 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{224629}+\frac{30649 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{224629}+\frac{10752 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{224629}+\frac{10752 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{224629}+\frac{10752 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{224629}+\frac{10752 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{224629}+\frac{10752 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{224629}+\frac{10752 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{224629}-\frac{3763 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{224629}-\frac{3763 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{224629}-\frac{3763 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{224629}-\frac{3763 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{224629}-\frac{3763 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{224629}-\frac{3763 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{224629}+\left(\left\{\begin{array}{cc}1 & 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) = 60
a(6) = 191
a(n+2) = -1/3*a(n)-2/3*a(n+1)+1/3*a(n+4)+a(n+5)-1/3*a(n+6), 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) = 60
a \! \left(6\right) = 191
a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{3}-\frac{2 a \! \left(n +1\right)}{3}+\frac{a \! \left(n +4\right)}{3}+a \! \left(n +5\right)-\frac{a \! \left(n +6\right)}{3}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16435/
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[20,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[19,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[4,x]*F[7,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[13,x]+F[22,x]+F[63,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[36,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[18,x]+F[7,x]
F[29,x] = F[25,x]+F[30,x]
F[30,x] = 2*F[13,x]+F[31,x]+F[35,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[18,x]
F[34,x] = F[35,x]
F[35,x] = F[25,x]*F[4,x]
F[36,x] = F[37,x]+F[50,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[2,x]*F[4,x]
F[43,x] = F[37,x]+F[44,x]
F[44,x] = 2*F[13,x]+F[45,x]+F[49,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[41,x]
F[48,x] = F[49,x]
F[49,x] = F[37,x]*F[4,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[21,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[21,x]*F[4,x]
F[56,x] = F[50,x]+F[57,x]
F[57,x] = 3*F[13,x]+F[58,x]+F[62,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[54,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[50,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[40,x]+F[65,x]
F[65,x] = F[21,x]+F[66,x]
F[66,x] = 2*F[13,x]+F[55,x]+F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[47,x]+F[69,x]
F[69,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_{20}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{22}\! \left(x \right)+F_{63}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{36}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \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_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \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_{21}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{50}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 3 F_{13}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \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_{54}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{40}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{21}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{55}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{47}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \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_20(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_19(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_4(x)*F_7(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_13(x) + F_22(x) + F_63(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_36(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_18(x) + F_7(x))
Eq(F_29(x), F_25(x) + F_30(x))
Eq(F_30(x), 2*F_13(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_18(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_25(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_50(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_43(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_2(x)*F_4(x))
Eq(F_43(x), F_37(x) + F_44(x))
Eq(F_44(x), 2*F_13(x) + F_45(x) + F_49(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_41(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_37(x)*F_4(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_21(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_21(x)*F_4(x))
Eq(F_56(x), F_50(x) + F_57(x))
Eq(F_57(x), 3*F_13(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_54(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_50(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_40(x) + F_65(x))
Eq(F_65(x), F_21(x) + F_66(x))
Eq(F_66(x), 2*F_13(x) + F_55(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_47(x) + F_69(x))
Eq(F_69(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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 1], [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, 1, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"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, 2], [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": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 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, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 1], [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, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, 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