0132_0321_1230_2130_3021

Counting sequence:
1, 1, 2, 6, 19, 51, 123, 284, 651, 1505, 3523, 8338, 19891, 47695, 114715, 276392, 666571, 1608381, 3881923, 9370526, 22620947, 54610027, 131838203, 318283188, 768400843, 1855080601, 4478557187, 10812189482, 26102929971, 63018042503, 152139007259, 367296048448, 886731094667, 2140758227317, 5168247537795, 12477253290294, 30122754104595, 72722761484451, 175568277057147, 423859315581004, 1023286908199947, 2470433131960145, 5964153172097859, 14398739476131778, 34761632124335539, 83922003724775103, 202605639573856027, 489133282872455384, 1180872205318732875, 2850877693509884973, 6882627592338464323, 16616132878186772686, 40114893348711966227, 96845919575610659035, 233806732499933235451, 564459384575477078244, 1362725501650887337291, 3289910387877251695113, 7942546277405390666627, 19175002942688032964186, 46292552162781456527411, 111760107268250945947895, 269812766699283348348443, 651385640666817642566256, 1572584048032918633398539, 3796553736732654909276901, 9165691521498228451861763, 22127936779729111812905574, 53421565080956452077573651, 128971066941642015967949075, 311363698964240484013363323, 751698464870122983994562428, 1814760628704486452002369931, 4381219722279095887999178945, 10577200073262678228000599235, 25535619868804452344000243442, 61648439810871582916000946611, 148832499490547618176001991471, 359313438791966819268004778523, 867459377074481256712011391496, 2094232192940929332692027398347, 5055923762956339922096066018717, 12206079718853609176884159259843, 29468083200663558275864384355838, 71142246120180725728612927782163, 171752575441025009733090239723851, 414647397002230745194793407026427, 1001047369445486500122677053565972, 2416742135893203745440147513940171, 5834531641231893991002972081220473, 14085805418356991727446091676147459, 34006142477945877445895155433273738, 82098090374248746619236402542445107, 198202323226443370684367960517905767, 478502736827135487987972323577989915, 1155207796880714346660312607673610144, 2788918330588564181308597538924925835, 6733044458057842709277507685523168341, 16255007246704249599863612909970959747, 39243058951466341909004733505464775574, 94741125149636933417873079920900188947

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

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

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

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

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

Explicit closed form in Maple syntax:
10+1/2*(2^(1/2)-1)^(-n)+1/2*(-1-2^(1/2))^(-n)-20/3*n+1/2*n^2+1/6*n^3-piecewise(n = 0,10,n = 1,4,n = 2,1,0)

Explicit closed form in latex syntax:
10+\frac{\left(\sqrt{2}-1\right)^{-n}}{2}+\frac{\left(-1-\sqrt{2}\right)^{-n}}{2}-\frac{20 n}{3}+\frac{n^{2}}{2}+\frac{n^{3}}{6}-\left(\left\{\begin{array}{cc}10 & n =0 \\ 4 & n =1 \\ 1 & n =2 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 123
a(7) = 284
a(8) = 651
a(n+2) = -1/3*n^3-n^2+a(n)+2*a(n+1)+43/3*n-18, 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) = 51
a \! \left(6\right) = 123
a \! \left(7\right) = 284
a \! \left(8\right) = 651
a \! \left(n +2\right) = -\frac{n^{3}}{3}-n^{2}+a \! \left(n \right)+2 a \! \left(n +1\right)+\frac{43 n}{3}-18, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19508/
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[12,x]*F[4,x]
F[4,x] = F[24,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[12,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[12,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[25,x]+F[35,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[16,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[25,x]
F[35,x] = F[18,x]+F[36,x]+F[66,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[53,x]
F[38,x] = F[39,x]+F[6,x]
F[39,x] = F[18,x]+F[40,x]+F[45,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[10,x]+F[42,x]
F[42,x] = F[18,x]+F[22,x]+F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[12,x]+F[42,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[44,x]+F[47,x]
F[47,x] = F[48,x]+F[50,x]
F[48,x] = F[49,x]
F[49,x] = x^2
F[50,x] = 2*F[18,x]+F[51,x]+F[52,x]
F[51,x] = F[12,x]*F[21,x]
F[52,x] = F[12,x]*F[47,x]
F[53,x] = F[35,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[18,x]+F[58,x]+F[64,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[10,x]+F[61,x]
F[61,x] = F[18,x]+F[40,x]+F[43,x]
F[62,x] = F[57,x]+F[63,x]
F[63,x] = F[55,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[25,x]+F[57,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[65,x]+F[68,x]
F[68,x] = F[69,x]+F[72,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[12,x]
F[72,x] = 2*F[18,x]+F[73,x]+F[75,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[42,x]
F[75,x] = F[12,x]*F[68,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{24}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{25}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{66}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{53}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = x^{2}
F_{50}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{53}\! \left(x \right) = F_{35}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{18}\! \left(x \right)+F_{58}\! \left(x \right)+F_{64}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{10}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{55}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{25}\! \left(x \right)+F_{57}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{72}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{42}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_24(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_12(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_12(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_25(x) + F_35(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_16(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_25(x))
Eq(F_35(x), F_18(x) + F_36(x) + F_66(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_53(x))
Eq(F_38(x), F_39(x) + F_6(x))
Eq(F_39(x), F_18(x) + F_40(x) + F_45(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_10(x) + F_42(x))
Eq(F_42(x), F_18(x) + F_22(x) + F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_12(x) + F_42(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_44(x) + F_47(x))
Eq(F_47(x), F_48(x) + F_50(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), x**2)
Eq(F_50(x), 2*F_18(x) + F_51(x) + F_52(x))
Eq(F_51(x), F_12(x)*F_21(x))
Eq(F_52(x), F_12(x)*F_47(x))
Eq(F_53(x), F_35(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_18(x) + F_58(x) + F_64(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_10(x) + F_61(x))
Eq(F_61(x), F_18(x) + F_40(x) + F_43(x))
Eq(F_62(x), F_57(x) + F_63(x))
Eq(F_63(x), F_55(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_25(x) + F_57(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_65(x) + F_68(x))
Eq(F_68(x), F_69(x) + F_72(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_12(x))
Eq(F_72(x), 2*F_18(x) + F_73(x) + F_75(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_42(x))
Eq(F_75(x), F_12(x)*F_68(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:
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