0213_0231_1032_1230_2130

Counting sequence:
1, 1, 2, 6, 19, 56, 155, 410, 1051, 2636, 6511, 15910, 38583, 93072, 223699, 536370, 1284147, 3071892, 7346007, 17567294, 42021839, 100562392, 240787323, 576894154, 1383041291, 3317810908, 7964137151, 19128644694, 45969768295, 110531310368, 265891538339, 639898467938, 1540588200547, 3710337450788, 8938714523815, 21540521858158, 51920973992319, 125176311412584, 301844100087883, 727971158390778, 1755930990997563, 4235984849688684, 10219929231073551, 24659350637419142, 59504545645450775, 143598073184143024, 346555556478870387, 836410118979128722, 2018760067925573267, 4872663617435077300, 11761483659261154295, 28390282911399883422, 68531071957969205551, 165429908829201442040, 399344727720191542043, 964024788677409745706, 2327200650291034102123, 5618029772492269346300, 13562449547342646105439, 32741271542514689214646, 79041605925452241921735, 190817565864391531976256, 460662614365803871998787, 1112113971558384104799426, 2684831758485837132401859, 6481657584693579257517636, 15647902508514018995878215, 37776964539631627091380430, 91200817016853219153300191, 220176532538001809928199816, 531549676235208033231931371, 1283267325719167675160112730, 3098066913947137801735431323, 7479365735864821756291872652, 18056726369587917552229667759, 43592672081679687901749583590, 105241773023858872568080372087, 254075613666487625723322974672, 613391772541548577906970760467, 1480856665340081826364591664114, 3575100040844842594376485006579, 8631046471160300290769576674772, 20837172129195054823562374674583, 50305348417146723425873211307774, 121447783129755309036637393151631, 293200740591339316992546839921688, 707848911306094615550012058791611, 1708897847519443336232099531448714, 4125643155633997750459201474277835, 9960181218679845534372349594584348, 24045999635407249758311327711411199, 58052168419578962018537324750945558, 140150312025249397909125547955598887, 338352742952476186421577424697174368, 816855697657058588636478130520885475, 1972063935244426920892168602282572578, 4760983157149819387674564068576788259, 11494029417648359296465752007224670564, 27749040308848084552488894220217184743, 66992106628533540288077023925269175598, 161733246673065026387645571432431856703

Generating function in Maple syntax:
-(x^5+4*x^4+6*x^3-12*x^2+6*x-1)/(x-1)/(x^2+2*x-1)/(2*x-1)^2

Generating function in latex syntax:
-\frac{x^{5}+4 x^{4}+6 x^{3}-12 x^{2}+6 x -1}{\left(x -1\right) \left(x^{2}+2 x -1\right) \left(2 x -1\right)^{2}}

Generating function in sympy syntax:
(-x**5 - 4*x**4 - 6*x**3 + 12*x**2 - 6*x + 1)/((x - 1)*(2*x - 1)**2*(x**2 + 2*x - 1))

Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+2*x-1)*(2*x-1)^2*F(x)+x^5+4*x^4+6*x^3-12*x^2+6*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+2 x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{5}+4 x^{4}+6 x^{3}-12 x^{2}+6 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/4*(2-2^(1/2))*(-1-2^(1/2))^(-n)+1/4*(2+2^(1/2))*(2^(1/2)-1)^(-n)+2+1/4*(n-7)*2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(2-\sqrt{2}\right) \left(-1-\sqrt{2}\right)^{-n}}{4}+\frac{\left(2+\sqrt{2}\right) \left(\sqrt{2}-1\right)^{-n}}{4}+2+\frac{\left(n -7\right) 2^{n}}{4} & \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(n+4) = 4*a(n)+4*a(n+1)-11*a(n+2)+6*a(n+3)-4, 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) = 19
a \! \left(5\right) = 56
a \! \left(n +4\right) = 4 a \! \left(n \right)+4 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)-4, \quad n \geq 6

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21844/
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[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[4,x]*F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = F[4,x]*F[8,x]
F[10,x] = F[11,x]+F[41,x]
F[11,x] = F[12,x]+F[29,x]
F[12,x] = F[13,x]+F[19,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[4,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[16,x]*F[4,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[1,x]+F[23,x]+F[25,x]
F[23,x] = F[24,x]
F[24,x] = F[16,x]*F[22,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[16,x]*F[27,x]*F[4,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[20,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]*F[8,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[39,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[29,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[14,x]*F[16,x]*F[4,x]
F[38,x] = F[14,x]*F[17,x]
F[39,x] = F[10,x]*F[40,x]
F[40,x] = F[9,x]
F[41,x] = F[19,x]+F[38,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_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{41}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{29}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \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_{4}\! \left(x \right)
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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{16}\! \left(x \right) F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{9}\! \left(x \right)
F_{41}\! \left(x \right) = F_{19}\! \left(x \right)+F_{38}\! \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_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_10(x)*F_4(x)*F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), F_4(x)*F_8(x))
Eq(F_10(x), F_11(x) + F_41(x))
Eq(F_11(x), F_12(x) + F_29(x))
Eq(F_12(x), F_13(x) + F_19(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)*F_4(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_16(x)*F_4(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_1(x) + F_23(x) + F_25(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_16(x)*F_22(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_16(x)*F_27(x)*F_4(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_20(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x)*F_8(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_39(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_29(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_14(x)*F_16(x)*F_4(x))
Eq(F_38(x), F_14(x)*F_17(x))
Eq(F_39(x), F_10(x)*F_40(x))
Eq(F_40(x), F_9(x))
Eq(F_41(x), F_19(x) + F_38(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": [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, 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": [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, 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": [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": false, "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, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": 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