0132_0321_1302_1320_2301

Counting sequence:
1, 1, 2, 6, 19, 55, 149, 388, 988, 2486, 6221, 15545, 38887, 97542, 245558, 620712, 1575631, 4016083, 10276025, 26386024, 67964560, 175541594, 454465937, 1178943341, 3063489739, 7971680010, 20767664234, 54155151948, 141328692883, 369059174791, 964238218781, 2520300038380, 6589682574532, 17234252170622, 45083009166293, 117952498302497, 308645636722351, 807718123892238, 2113958979140510, 5533024942163184, 14482779385140055, 37910502849885499, 99238833559866497, 259785656864600176, 680076355592078584, 1780357648004669090, 4660820666561484569, 12201743711865874133, 31943671597222272403, 83627758151801123346, 218936506633437280466, 573175428561534728340, 1500576831202238214427, 3928528606397369113327, 10284954944794340679653, 26926225889794782348532, 70493497544608637841196, 184553807376869139384518, 483166987837276287249245, 1264945246608717717272969, 3311664860878798816461175, 8670041409692334560037654, 22698443227297140367794182, 59425255413936205248206136, 155577256145064208179699679, 407306376976518875682949795, 1066341598083331313225875529, 2791717854647780815853352568, 7308810822161878564341982048, 19134712287548101589769090026, 50095321318115943335320074401, 131151242074492810087224292925, 343358385425600745089066297947, 898923874652490131146695936282, 2353413158251639439567052878138, 6161315437180784528551702825692, 16130532822725060345212890640291, 42230282360418355939597348750423, 110560312898488460407121334066605, 289450653577184999285893848651004, 757791642241784621732900278870420, 1983924261814489307025658475089934, 5193981120232092726666120826982117, 13598019052338144817809480779668721, 35600075942486127796821245884942399, 93202208584109959073542847271582750, 244006549422987487147125960023829614, 638817438901468571257555327590304128, 1672445765695307551291146545532579943, 4378519854973547105719429220987832331, 11463113792726148559618894674210514961, 30010821510051785655730089380840517760, 78569350710813497562935697557979867592, 205697230568538315324162961334987181362, 538522340885862724952439722255858744873, 1409869791868696532540358516500544941477, 3691087034274568458525898928282091359011, 9663391310053788494437581428219166700578, 25299086894064549286958765591723897885090, 66233869368455749809525423648852733264164, 173402521203855252865276657621041170575915

Generating function in Maple syntax:
-(2*x^7-6*x^6+9*x^5-25*x^4+37*x^3-25*x^2+8*x-1)/(x^2-3*x+1)/(x-1)^2/(2*x-1)^2

Generating function in latex syntax:
-\frac{2 x^{7}-6 x^{6}+9 x^{5}-25 x^{4}+37 x^{3}-25 x^{2}+8 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{2}}

Generating function in sympy syntax:
(-2*x**7 + 6*x**6 - 9*x**5 + 25*x**4 - 37*x**3 + 25*x**2 - 8*x + 1)/((x - 1)**2*(2*x - 1)**2*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^2*(2*x-1)^2*F(x)+2*x^7-6*x^6+9*x^5-25*x^4+37*x^3-25*x^2+8*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{2} F \! \left(x \right)+2 x^{7}-6 x^{6}+9 x^{5}-25 x^{4}+37 x^{3}-25 x^{2}+8 x -1 = 0

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

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \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{2^{n} n}{4}-n -\\2^{-1+n}+2 & \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) = 55
a(6) = 149
a(7) = 388
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)+n-3, 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) = 55
a \! \left(6\right) = 149
a \! \left(7\right) = 388
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+n -3, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21754/
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[12,x]*F[17,x]*F[8,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]*F[8,x]
F[20,x] = F[21,x]+F[23,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[22,x]*F[8,x]
F[25,x] = F[13,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[30,x]
F[27,x] = 0
F[28,x] = F[29,x]*F[8,x]
F[29,x] = F[23,x]+F[26,x]
F[30,x] = F[25,x]*F[8,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_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \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_12(x)*F_17(x)*F_8(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x)*F_8(x))
Eq(F_20(x), F_21(x) + F_23(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_22(x)*F_8(x))
Eq(F_25(x), F_13(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_30(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_29(x)*F_8(x))
Eq(F_29(x), F_23(x) + F_26(x))
Eq(F_30(x), F_25(x)*F_8(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 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], [1, 0], [1, 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, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 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