0123_0132_1203_2103_2130

Counting sequence:
1, 1, 2, 6, 19, 53, 144, 379, 975, 2466, 6159, 15241, 37460, 91615, 223251, 542598, 1316243, 3188581, 7716672, 18661883, 45108831, 108996258, 263299983, 635933465, 1535738212, 3708375743, 8954119587, 21619360710, 52197459475, 126022036757, 304254544176, 734552921851, 1773396859887, 4281407602914, 10336313855631, 24954205114537, 60245007080756, 145444690527391, 351135172243635, 847716338689734, 2046570015582611, 4940859965981509, 11928295914300000, 28797461688622715, 69523235688169599, 167843960222616162, 405211201090497807, 978266436788168249, 2361744197681110468, 5701755035491651967, 13765254604635007299, 33232264799631006342, 80229785119909490707, 193691836551061820405, 467613460715559995088, 1128918762093953089147, 2725450991681299458063, 6579820756625324293410, 15885092523330256485135, 38350005833583788584393, 92585104240376666874644, 223520214396427350927199, 539625533168296448055123, 1302771280955185016520966, 3145168095444001808982419, 7593107472443803672494373, 18331383041318779939580544, 44255873556703484735628539, 106843130157390597160490655, 257942133875861539130661666, 622727397916300830166015119, 1503396929720262552341866841, 3629521257376193227454543140, 8762439444504431775777262655, 21154400146437203748163930851, 51071239737464382040852395078, 123296879621506267506860711443, 297664998980706982470469035221, 718626877583297430449931802992, 1734918754147920162667762649083, 4188464385880151161873667713071, 10111847525909883156468788047458, 24412159437702638384409841327503, 58936166401319617283065504482025, 142284492240349173829497924813812, 343505150882029921568501651922463, 829294794004428595473186233727987, 2002094738890919167038993244505670, 4833484271786319402974258307915155, 11669063282463643860717989847863557, 28171610836713747685739360308549536, 68012284955891369241215144766825851, 164196180748496862498651595190603967, 396404646452885709898111935549360354, 957005473654269289365178621531746831, 2310415593761425935758686764500356217, 5577836661177123854563621331398829956, 13466088916115678049578085464032088831, 32510014493408487155454435186780097155, 78486117902932664134156041781340291334, 189482250299273834669773952309939269651

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

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

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

Implicit equation for the generating function in Maple syntax:
(x^2+2*x-1)*(x^2+x-1)^2*F(x)+x^8+2*x^7-3*x^6-5*x^5+3*x^4+6*x^3-3*x+1 = 0

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

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/25*(-10*n+18*5^(1/2)+25)*(-1/2*5^(1/2)-1/2)^(-n)+1/25*(-10*n-18*5^(1/2)+25)*(1/2*5^(1/2)-1/2)^(-n)+(-1-2^(1/2))^(-n)+(2^(1/2)-1)^(-n))

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-10 n +18 \sqrt{5}+25\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{25}+\\\frac{\left(-10 n -18 \sqrt{5}+25\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{25}+\left(-1-\sqrt{2}\right)^{-n}+\left(\sqrt{2}-1\right)^{-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) = 53
a(6) = 144
a(7) = 379
a(8) = 975
a(n+6) = a(n)+4*a(n+1)+2*a(n+2)-6*a(n+3)-2*a(n+4)+4*a(n+5), n >= 9

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) = 53
a \! \left(6\right) = 144
a \! \left(7\right) = 379
a \! \left(8\right) = 975
a \! \left(n +6\right) = a \! \left(n \right)+4 a \! \left(n +1\right)+2 a \! \left(n +2\right)-6 a \! \left(n +3\right)-2 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16450/
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[17,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[16,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[4,x]
F[16,x] = F[4,x]*F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[13,x]+F[19,x]+F[30,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[13,x]+F[14,x]+F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[27,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[16,x]
F[27,x] = F[22,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[22,x]*F[4,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[2,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[2,x]*F[4,x]
F[35,x] = F[18,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[18,x]*F[4,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_{17}\! \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_{16}\! \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_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \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_17(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_16(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_4(x)*F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_13(x) + F_19(x) + F_30(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_13(x) + F_14(x) + F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_27(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_16(x))
Eq(F_27(x), F_22(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_22(x)*F_4(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_2(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_2(x)*F_4(x))
Eq(F_35(x), F_18(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_18(x)*F_4(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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 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": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 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], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [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, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 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, 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