0213_1032_1320_2130_3021

Counting sequence:
1, 1, 2, 6, 19, 54, 142, 360, 903, 2270, 5746, 14654, 37613, 97028, 251238, 652342, 1697273, 4422682, 11537492, 30123628, 78701557, 205717986, 537927032, 1407013266, 3681014139, 9631833144, 25206094732, 65969671610, 172669363143, 451971306110, 1183110334266, 3097091257664, 8107626563843, 21224714687638, 55564370010558, 145464100371382, 380819341163113, 996976743242332, 2610076528818482, 6833184123728718, 17889338403405877, 46834556208572946, 122613780466489312, 321005685679256676, 840401077548013809, 2200193148918261434, 5760169573113735172, 15080297978236885610, 39480689177224817783, 103361699184693374000, 270604267639366931736, 708450822258430692274, 1854747636185971704323, 4855790960399577557398, 12712622993212947260678, 33282073515639636830872, 87133588546506708466303, 228118674109481979059982, 597222397753142209722122, 1563548447092350612149358, 4093422799408721550839397, 10716719662903437888624756, 28056735612840839811577342, 73453486022697576939224486, 192303720149408881792364529, 503457669813843050010441674, 1318069280068748231384143212, 3450750151945657570432392860, 9034181138874736332493886573, 23651793190891575132211012690, 61921198286226036474462688656, 162111801372638629111824175234, 424414205241394040502304130803, 1111130813170951871677676856808, 2908978231910278333095903773348, 7615803877837516644740389187562, 19938433392157538635385973297343, 52199496279745633329938949782382, 136660055409300429491473714270162, 357780669872597791418567869536048, 936681954057377217312401247423867, 2452265191997302405614978578980166, 6420113621330067089725219902081622, 16808075670783973043946051952473446, 44004113388604000402883677605838049, 115204264490192324886246464166124364, 301608680072301567698938681494790602, 789621775707369565096735513522850110, 2067256647011121501363599725483060013, 5412148165248623686538727395745028930, 14169187848580007053341909927389526776, 37095415380181912463665657317698657044, 97117058291346760318012371888256765033, 254255759492620428451086078072172392618, 665650220184038644956675101778462039068, 1742694901054543746261797706163617098202, 4562434482969689073514434974513196127983, 11944608547834716433652941132977585160032, 31271391160494846146187256255622787234352, 81869564933570593842394563296297232445538, 214337303640058479055967904958081822049787

Generating function in Maple syntax:
(3*x^7-11*x^6+21*x^5-38*x^4+43*x^3-26*x^2+8*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^4

Generating function in latex syntax:
\frac{3 x^{7}-11 x^{6}+21 x^{5}-38 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}

Generating function in sympy syntax:
(3*x**7 - 11*x**6 + 21*x**5 - 38*x**4 + 43*x**3 - 26*x**2 + 8*x - 1)/((x - 1)**4*(2*x - 1)*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^4*F(x)-3*x^7+11*x^6-21*x^5+38*x^4-43*x^3+26*x^2-8*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 -1\right)^{4} F \! \left(x \right)-3 x^{7}+11 x^{6}-21 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(18*5^(1/2)-30)*(3/2-1/2*5^(1/2))^(-n)+1/30*(-18*5^(1/2)-30)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-n^2-1/6*n+2^(n-1)+1)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(18 \sqrt{5}-30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-18 \sqrt{5}-30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-n^{2}-\frac{n}{6}\\+2^{n -1}+1 & \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) = 54
a(6) = 142
a(7) = 360
a(n+3) = 1/6*n^3-n^2+2*a(n)-7*a(n+1)+5*a(n+2)-13/6*n+4, n >= 8

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) = 54
a \! \left(6\right) = 142
a \! \left(7\right) = 360
a \! \left(n +3\right) = \frac{n^{3}}{6}-n^{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{13 n}{6}+4, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20569/
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[16,x]*F[4,x]
F[4,x] = F[23,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[20,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[17,x]
F[20,x] = F[21,x]
F[21,x] = F[16,x]*F[22,x]
F[22,x] = F[9,x]
F[23,x] = F[10,x]+F[24,x]
F[24,x] = F[25,x]+F[26,x]+F[40,x]
F[25,x] = 0
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[28,x]+F[34,x]
F[28,x] = F[14,x]+F[29,x]
F[29,x] = F[25,x]+F[30,x]+F[31,x]
F[30,x] = F[16,x]*F[28,x]
F[31,x] = F[16,x]*F[32,x]
F[32,x] = F[14,x]+F[33,x]
F[33,x] = F[31,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[25,x]+F[26,x]+F[36,x]
F[36,x] = F[16,x]*F[37,x]
F[37,x] = F[10,x]+F[35,x]
F[38,x] = F[39,x]
F[39,x] = F[16,x]*F[34,x]
F[40,x] = F[16,x]*F[41,x]
F[41,x] = F[23,x]+F[42,x]
F[42,x] = F[43,x]+F[49,x]
F[43,x] = F[44,x]
F[44,x] = F[16,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[14,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[16,x]*F[46,x]
F[49,x] = F[50,x]
F[50,x] = F[16,x]*F[51,x]
F[51,x] = F[42,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{23}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{40}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{34}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{16}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{10}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right) F_{34}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{23}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{16}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{42}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_23(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_16(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_20(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_17(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_16(x)*F_22(x))
Eq(F_22(x), F_9(x))
Eq(F_23(x), F_10(x) + F_24(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_40(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_34(x))
Eq(F_28(x), F_14(x) + F_29(x))
Eq(F_29(x), F_25(x) + F_30(x) + F_31(x))
Eq(F_30(x), F_16(x)*F_28(x))
Eq(F_31(x), F_16(x)*F_32(x))
Eq(F_32(x), F_14(x) + F_33(x))
Eq(F_33(x), F_31(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_25(x) + F_26(x) + F_36(x))
Eq(F_36(x), F_16(x)*F_37(x))
Eq(F_37(x), F_10(x) + F_35(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_16(x)*F_34(x))
Eq(F_40(x), F_16(x)*F_41(x))
Eq(F_41(x), F_23(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_49(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_16(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_14(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_16(x)*F_46(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_16(x)*F_51(x))
Eq(F_51(x), F_42(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 2, 1, 3], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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