0132_0213_0231_1320_2301

Counting sequence:
1, 1, 2, 6, 19, 54, 138, 333, 791, 1898, 4651, 11645, 29657, 76411, 198289, 516699, 1349468, 3528653, 9232581, 24164158, 63253756, 165589565, 433505762, 1134916666, 2971231034, 7778760794, 20365032948, 53316316548, 139583891721, 365435329768, 956722064436, 2504730825635, 6557470369317, 17167680233396, 44945570275629, 117669030531339, 308061521248703, 806515533136893, 2111485078075211, 5527939700992353, 14472334024795066, 37889062373274855, 99194853094899447, 259695496911280476, 679891637638785074, 1779979416004902959, 4660046610375736108, 12200160415122100688, 31940434634990343176, 83621143489848686788, 218922995834555454646, 573147844013817392850, 1500520536206896416527, 3928413764606871524906, 10284720757613717800494, 26925748508234281491529, 70492524767089126260163, 184551825793033096844558, 483162952612010163796991, 1264937032042997394036073, 3311648143516982017765301, 8670007398507948658676495, 22698374052006863957641557, 59425114757512643213584311, 155576970220531065682404264, 407305795904080553832876049, 1066340417491710595815423993, 2791715456571051233612546378, 7308805952221443105021313656, 19134702400093278081450438833, 50095301248058391139328990406, 131151201344081895336535460790, 343358302784187294870276258662, 898923707008479989274292117566, 2353412818241252672952598829384, 6161314747715278029583503036144, 16130531424904581415797908871973, 42230279526998466217810222097148, 110560307156090817237632755858296, 289450641941273985495088043834943, 757791618667731139247631373918961, 1983924214061919432247806076106360, 5193981023518027157495786852493217, 13598018856492162040239554479371671, 35600075545958458963222876583521979, 93202207781383214849429075268992689, 244006547798191185585064349221149103, 638817435613190341905763972392038493, 1672445759041379840132227567952437286, 4378519841510949178490918731462627403, 11463113765491467695340528626432678091, 30010821454963453907530667147832515080, 78569350599398894027251472817061846222, 205697230343233228174223751303349869251, 538522340430300790495419781092984469424, 1409869790947669143312035591975600104684, 3691087032412706639440686994833812263508, 9663391306290450775010025392525832953288, 25299086886458645685589389182743682707626, 66233869353085486281758142155705211119838, 173402521172797813159685037284371946436171

Generating function in Maple syntax:
(x^8-3*x^7-2*x^6+13*x^5-23*x^4+29*x^3-20*x^2+7*x-1)/(x^2-3*x+1)/(-1+x)^5

Generating function in latex syntax:
\frac{x^{8}-3 x^{7}-2 x^{6}+13 x^{5}-23 x^{4}+29 x^{3}-20 x^{2}+7 x -1}{\left(x^{2}-3 x +1\right) \left(-1+x \right)^{5}}

Generating function in sympy syntax:
(x**8 - 3*x**7 - 2*x**6 + 13*x**5 - 23*x**4 + 29*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**5*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(-1+x)^5*F(x)+x^8-3*x^7-2*x^6+13*x^5-23*x^4+29*x^3-20*x^2+7*x-1 = 0

Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(-1+x \right)^{5} F \! \left(x \right)+x^{8}-3 x^{7}-2 x^{6}+13 x^{5}-23 x^{4}+29 x^{3}-20 x^{2}+7 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/120*(-12*5^(1/2)+60)*(3/2-1/2*5^(1/2))^(-n)+1/120*(12*5^(1/2)+60)*(3/2+1/2*5^(1/2))^(-n)+1/24*n^4+1/4*n^3-61/24*n^2+25/4*n-5)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-12 \sqrt{5}+60\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(12 \sqrt{5}+60\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}+\frac{n^{3}}{4}-\\\frac{61 n^{2}}{24}+\frac{25 n}{4}-5 & \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) = 138
a(7) = 333
a(8) = 791
a(n+2) = -a(n)+3*a(n+1)-1/24*(-1+n)*(n+1)*(n^2+10*n-48), 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) = 54
a \! \left(6\right) = 138
a \! \left(7\right) = 333
a \! \left(8\right) = 791
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{\left(-1+n \right) \left(n +1\right) \left(n^{2}+10 n -48\right)}{24}, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21584/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[21,x]+F[8,x]
F[8,x] = F[11,x]*F[9,x]
F[9,x] = F[10,x]+F[19,x]
F[10,x] = F[11,x]*F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[14,x]*F[9,x]
F[19,x] = F[20,x]
F[20,x] = F[11,x]^2*F[14,x]*F[17,x]
F[21,x] = F[22,x]
F[22,x] = F[11,x]*F[14,x]*F[23,x]
F[23,x] = F[24,x]+F[33,x]
F[24,x] = F[25,x]+F[27,x]
F[25,x] = F[11,x]*F[26,x]
F[26,x] = F[11,x]+F[12,x]
F[27,x] = F[11,x]*F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[11,x]*F[14,x]*F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[14,x]*F[32,x]
F[32,x] = F[1,x]+F[31,x]
F[33,x] = F[26,x]*F[30,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_{14}\! \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_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{21}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{19}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{33}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{12}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right) F_{30}\! \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_14(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_14(x)*F_7(x))
Eq(F_7(x), F_21(x) + F_8(x))
Eq(F_8(x), F_11(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_19(x))
Eq(F_10(x), F_11(x)*F_15(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_14(x)*F_9(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_11(x)**2*F_14(x)*F_17(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_11(x)*F_14(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_33(x))
Eq(F_24(x), F_25(x) + F_27(x))
Eq(F_25(x), F_11(x)*F_26(x))
Eq(F_26(x), F_11(x) + F_12(x))
Eq(F_27(x), F_11(x)*F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_11(x)*F_14(x)*F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_14(x)*F_32(x))
Eq(F_32(x), F_1(x) + F_31(x))
Eq(F_33(x), F_26(x)*F_30(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 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