0132_0213_0321_1032_1320_2301

Counting sequence:
1, 1, 2, 6, 18, 46, 113, 278, 693, 1751, 4471, 11501, 29734, 77137, 200590, 522510, 1362756, 3557464, 9293135, 24289040, 63508299, 166104617, 434543221, 1137000551, 2975409628, 7787130931, 20381788588, 53349845928, 139650971658, 365569514266, 956990461901, 2505267653306, 6558544062129, 17169827661707, 44949865180675, 117677620396145, 308078701039906, 806549892788389, 2111553797455450, 5528077139838930, 14472608902583904, 37889612128958572, 99195952606384091, 259697695934378996, 679896035685124263, 1779988212097737341, 4660064202561575713, 12200195599493966603, 31940505003734278648, 83621284227336779431, 218923277309531880856, 573148406963770506636, 1500521662106802927174, 3928416016406684852302, 10284725261213344785785, 26925757515433535818430, 70492542781487635297581, 184551861821830115331839, 483163024669604201214415, 1264937176158185469345845, 3311648431747358168893534, 8670007974968700961477177, 22698375204928368563824486, 59425117063355652426570966, 155576974832217084109039548, 407305805127452590686851776, 1066340435938454669524125863, 2791715493464539381030747928, 7308806026008419399858564163, 19134702547667230671125839121, 50095301543206296318680744461, 131151201934377705695239978991, 343358303964778915587686364244, 898923709369663230709113459547, 2353412822963619155822242708420, 6161314757160010995322792056240, 16130531443794047347276488243906, 42230279564777398080767382245314, 110560307231648680963547077634405, 289450642092389712946916688945410, 757791618969962594151288665779689, 1983924214666382342055120661552307, 5193981024726952977110416025197531, 13598018858910013679468812826683961, 35600075550794162241681393280144858, 93202207791054621406346108664334861, 244006547817533998698898416014031538, 638817435651875968133432105980106778, 1672445759118751092587563835130986328, 4378519841665691683401591265822250836, 11463113765800952705161873695154567091, 30010821455582423927173357285279055996, 78569350600636834066536853091957815839, 205697230345709108252794511853144825317, 538522340435252550652561302192577531705, 1409869790957572663626318634174789517075, 3691087032432513680069253079232194518256, 9663391306330064856267157561322601039439, 25299086886537873848103653520337222607920, 66233869353243942606786670830892294804500, 173402521173114725809742094634746117850494

Generating function in Maple syntax:
(2*x^8-10*x^7+14*x^6-7*x^5+13*x^4-24*x^3+19*x^2-7*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^3

Generating function in latex syntax:
\frac{2 x^{8}-10 x^{7}+14 x^{6}-7 x^{5}+13 x^{4}-24 x^{3}+19 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(2*x**8 - 10*x**7 + 14*x**6 - 7*x**5 + 13*x**4 - 24*x**3 + 19*x**2 - 7*x + 1)/((x - 1)**3*(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)^3*F(x)+2*x^8-10*x^7+14*x^6-7*x^5+13*x^4-24*x^3+19*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 -1\right)^{3} F \! \left(x \right)+2 x^{8}-10 x^{7}+14 x^{6}-7 x^{5}+13 x^{4}-24 x^{3}+19 x^{2}-7 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/20*(-2*5^(1/2)+10)*(3/2-1/2*5^(1/2))^(-n)+1/20*(2*5^(1/2)+10)*(3/2+1/2*5^(1/2))^(-n)+1/2*n^2-3/2*n+1/4*2^n-1)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-2 \sqrt{5}+10\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(2 \sqrt{5}+10\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{n^{2}}{2}-\frac{3 n}{2}+\frac{2^{n}}{4}\\-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) = 18
a(5) = 46
a(6) = 113
a(7) = 278
a(8) = 693
a(n+3) = 1/2*n^2+2*a(n)-7*a(n+1)+5*a(n+2)-3/2*n-3, 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) = 18
a \! \left(5\right) = 46
a \! \left(6\right) = 113
a \! \left(7\right) = 278
a \! \left(8\right) = 693
a \! \left(n +3\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{3 n}{2}-3, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22228/
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[8,x]
F[4,x] = F[15,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[8,x]*F[9,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[8,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[18,x]+F[23,x]
F[18,x] = F[19,x]+F[21,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[5,x]+F[6,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]^2*F[6,x]*F[8,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[8,x]
F[26,x] = F[27,x]+F[38,x]
F[27,x] = F[28,x]+F[30,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[29,x]*F[8,x]
F[32,x] = F[13,x]+F[33,x]
F[33,x] = F[34,x]+F[35,x]+F[37,x]
F[34,x] = 0
F[35,x] = F[36,x]*F[8,x]
F[36,x] = F[30,x]+F[33,x]
F[37,x] = F[32,x]*F[8,x]
F[38,x] = F[13,x]*F[29,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{15}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{5}\! \left(x \right)+F_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12} \left(x \right)^{2} F_{6}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{8}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{38}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = 0
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right) F_{8}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \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_8(x))
Eq(F_4(x), F_15(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_8(x)*F_9(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_8(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_18(x) + F_23(x))
Eq(F_18(x), F_19(x) + F_21(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_5(x) + F_6(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)**2*F_6(x)*F_8(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_8(x))
Eq(F_26(x), F_27(x) + F_38(x))
Eq(F_27(x), F_28(x) + F_30(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_29(x)*F_8(x))
Eq(F_32(x), F_13(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_35(x) + F_37(x))
Eq(F_34(x), 0)
Eq(F_35(x), F_36(x)*F_8(x))
Eq(F_36(x), F_30(x) + F_33(x))
Eq(F_37(x), F_32(x)*F_8(x))
Eq(F_38(x), F_13(x)*F_29(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, 3, 2, 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, 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, 3, 2, 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, 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, 3, 2, 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, 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"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [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, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], 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