0213_0231_1032_1230_1302

Counting sequence:
1, 1, 2, 6, 19, 56, 157, 428, 1147, 3039, 7989, 20889, 54421, 141444, 367084, 951906, 2467604, 6396697, 16585826, 43021802, 111648988, 289909147, 753221541, 1958124034, 5093429756, 13256360215, 34519948848, 89936075135, 234423253430, 611300958905, 1594710525163, 4161656642727, 10864108733713, 28369527964633, 74101640872690, 193601950567378, 505926258068832, 1322362724274608, 3456933488162035, 9038585526531850, 23635874844477610, 61815601547322866, 161686524260035337, 422954406414709571, 1106502824389391954, 2894986079556425746, 7574807430036837367, 19820949963577811168, 51868302904963803657, 135738047001638680404, 355239060531044700299, 929730815459645677379, 2433375931102994902281, 6369054192834461979109, 16670664319763751578669, 43635678772931824216296, 114219491636915403625564, 298983548207279104580762, 782639900916824475485536, 2048723995823457755214917, 5363038830571201658395198, 14039245721256887485441362, 36752032218429451215352176, 96210652594775233598467899, 251865515426573559020810341, 659352392677798322721110430, 1726113779218926186256604008, 4518807882078580099890027075, 11829888934860884993111659040, 30969880350002448494261375995, 81077477162159551291147605398, 212257262416575825138833973017, 555682015113765022826501652223, 1454760200194976485546930244323, 3808532137919345772431719066725, 9970681740212695841171734427593, 26103153972680358102890583313142, 68337945342139772359822082631898, 178908741283954809993611884792920, 468383766742821573307806616380140, 1226232070309901416985479563237095, 3210288060971521354733182677524606, 8404575428309776852591341481932922, 22003306448567034517155573902888886, 57605037575988949997742266622225889, 150811094120470423130205064718925811, 394826589213630269432426144794180702, 1033664824776664939997873713628897158, 2706158937860081516071706398826582667, 7084791188931106532012898153586835144, 18548166275045943848936055479048415109, 48559595226956134187166340328982795244, 127130358085818059110623026946317734891, 332830871534789246753288503082764643399, 871360844261562530331608029051498191629, 2281248378149112144930235380338226092561, 5972376657899204931690481409286620562149, 15635863852628429219483183172608890199452, 40935173652688310660696641243222236067996, 107169561217076957412338328523854921377282, 280573287085099782145722804115659948748908

Generating function in Maple syntax:
(2*x^6-6*x^5+10*x^4-20*x^3+18*x^2-7*x+1)/(2*x-1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)

Generating function in latex syntax:
\frac{2 x^{6}-6 x^{5}+10 x^{4}-20 x^{3}+18 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}

Generating function in sympy syntax:
(2*x**6 - 6*x**5 + 10*x**4 - 20*x**3 + 18*x**2 - 7*x + 1)/((2*x - 1)*(x**2 - 3*x + 1)*(x**3 - 2*x**2 + 3*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x^3-2*x^2+3*x-1)*F(x)-2*x^6+6*x^5-10*x^4+20*x^3-18*x^2+7*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)-2 x^{6}+6 x^{5}-10 x^{4}+20 x^{3}-18 x^{2}+7 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/13800*(23*2^(1/3)*((-29/23*23^(1/2)+I)*3^(1/2)-87/23*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+4600-575*((1/23*23^(1/2)+I)*3^(1/2)-3/23*I*23^(1/2)-1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*(575*2^(2/3)*((-1/23*23^(1/2)+I)*3^(1/2)-3/23*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600-23*2^(1/3)*((29/23*23^(1/2)+I)*3^(1/2)-87/23*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*((50*3^(1/2)*2^(2/3)*23^(1/2)-1150*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600+(58*23^(1/2)*3^(1/2)*2^(1/3)-46*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/5*5^(1/2)*(3/2-1/2*5^(1/2))^(-n)-1/5*5^(1/2)*(3/2+1/2*5^(1/2))^(-n)-2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(23 \,2^{\frac{1}{3}} \left(\left(-\frac{29 \sqrt{23}}{23}+\mathrm{I}\right) \sqrt{3}-\frac{87 \,\mathrm{I} \sqrt{23}}{23}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+4600-575 \left(\left(\frac{\sqrt{23}}{23}+\mathrm{I}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{23}}{23}-1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}\\+\\\frac{\left(575 \,2^{\frac{2}{3}} \left(\left(-\frac{\sqrt{23}}{23}+\mathrm{I}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{23}}{23}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600-23 \,2^{\frac{1}{3}} \left(\left(\frac{29 \sqrt{23}}{23}+\mathrm{I}\right) \sqrt{3}-\frac{87 \,\mathrm{I} \sqrt{23}}{23}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}\\+\\\frac{\left(\left(50 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{23}-1150 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600+\left(58 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}-46 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{13800}\\+\frac{\sqrt{5}\, \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{5}-\frac{\sqrt{5}\, \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5}-2^{n} & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 157
a(n+6) = -2*a(n)+11*a(n+1)-25*a(n+2)+34*a(n+3)-24*a(n+4)+8*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) = 56
a \! \left(6\right) = 157
a \! \left(n +6\right) = -2 a \! \left(n \right)+11 a \! \left(n +1\right)-25 a \! \left(n +2\right)+34 a \! \left(n +3\right)-24 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21841/
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[15,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[15,x]*F[7,x]
F[7,x] = F[26,x]+F[8,x]
F[8,x] = F[19,x]+F[9,x]
F[9,x] = F[10,x]*F[16,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]*F[15,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[13,x]*F[15,x]
F[15,x] = x
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[10,x]*F[15,x]*F[16,x]
F[19,x] = F[13,x]*F[20,x]*F[22,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[15,x]*F[20,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[13,x]*F[15,x]*F[25,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[13,x]^2*F[2,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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{26}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{19}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \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_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right) F_{20}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13} \left(x \right)^{2} F_{2}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_15(x)*F_7(x))
Eq(F_7(x), F_26(x) + F_8(x))
Eq(F_8(x), F_19(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_16(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_13(x)*F_15(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_13(x)*F_15(x))
Eq(F_15(x), x)
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_10(x)*F_15(x)*F_16(x))
Eq(F_19(x), F_13(x)*F_20(x)*F_22(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_15(x)*F_20(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_13(x)*F_15(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_13(x)**2*F_2(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 2, 1, 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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 2, 1, 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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], 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