0123_0213_0321_1032_1230

Counting sequence:
1, 1, 2, 6, 19, 51, 133, 341, 863, 2163, 5381, 13309, 32767, 80379, 196597, 479717, 1168319, 2840931, 6899333, 16737901, 40571743, 98274603, 237907381, 575662229, 1392377567, 3366708819, 8138378117, 19668630877, 47525971519, 114821237211, 277369772533, 669963435461, 1618101949823, 3907777947843, 9436879070981, 22787978540749, 55025721054367, 132865190453259, 320807641568437, 774583552805237, 1870180905609119, 4515357680883891, 10901720901097733, 26320448750521021, 63545916937023103, 153418879694333883, 370396870465224181, 894239008903848869, 2158927664831055167, 5212199891682225699, 12583538554428039557, 30379699213003370797, 73343781405364913119, 177068950873593460971, 427485060852272362933, 1032045827977579242581, 2491590227606312959583, 6015253304787969384723, 14522150880377780174981, 35059663151934586626589, 84641693357029067211967, 204343482211556948618139, 493329522471271419583477, 1191004256536356698055557, 2875341494308498636235519, 6941694162682381611608451, 16758743654731317141616133, 40459209142261126459168141, 97677217279485791188607263, 235813754381697151093692363, 569304947403808977890611381, 1374424091911172875904153909, 3318154016669870267757396767, 8010733896138344487535902579, 19339625350721421395063112197, 46689991681130911582129947517, 112719622880082693168258648319, 272129265775495195136518526331, 656978211099470877877038265333, 1586085801311232539762080185701, 3829150040395527135144168894143, 9244386335449469165536358488803, 22317923617988830177188766901381, 53880235384815858941857654350829, 130078398014398006904791599721567, 314037038667166790439215902031019, 758152489855841423158773500257717, 1830342047393069307507863095494677, 4418836642670419379676700077143519, 10668015448790786749865664021574611, 25754867772365750245416829663878533, 62177751457749801972716926436503261, 150110371616320383654885888711228223, 362398496547300628210559116207646043, 874907368424741757932144945823892981, 2112213240824424379787130657250177349, 5099333864928870988930969559113738367, 12310881000392727300498196373056635459, 29721095925135447475625615500384972037, 71753072969505866023145933764142505037, 173227242101831667064710495809301833119

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

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

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

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

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

Explicit closed form in Maple syntax:
-5/2*(2^(1/2)-1)^(1-n)-3/4*2^n-5/2*(-1-2^(1/2))^(1-n)-13/2*(2^(1/2)-1)^(-n)-13/2*(-1-2^(1/2))^(-n)+7/2*(2^(1/2)-1)^(-n-1)+7/2*(-1-2^(1/2))^(-n-1)-piecewise(n = 0,-11/4,n = 1,1/2,n = 3,1,0)

Explicit closed form in latex syntax:
-\frac{5 \left(\sqrt{2}-1\right)^{1-n}}{2}-\frac{3 \,2^{n}}{4}-\frac{5 \left(-1-\sqrt{2}\right)^{1-n}}{2}-\frac{13 \left(\sqrt{2}-1\right)^{-n}}{2}-\frac{13 \left(-1-\sqrt{2}\right)^{-n}}{2}+\frac{7 \left(\sqrt{2}-1\right)^{-n -1}}{2}+\frac{7 \left(-1-\sqrt{2}\right)^{-n -1}}{2}-\left(\left\{\begin{array}{cc}-\frac{11}{4} & n =0 \\ \frac{1}{2} & n =1 \\ 1 & n =3 \\ 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) = 133
a(n+3) = -2*a(n)-3*a(n+1)+4*a(n+2), n >= 7

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) = 133
a \! \left(n +3\right) = -2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right), \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16819/
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[20,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[15,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[69,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[35,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[22,x]+F[27,x]+F[34,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = x^2
F[32,x] = F[33,x]
F[33,x] = F[15,x]*F[4,x]
F[34,x] = F[19,x]*F[4,x]
F[35,x] = F[36,x]+F[59,x]
F[36,x] = F[22,x]+F[23,x]+F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[46,x]
F[42,x] = F[4,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[4,x]
F[46,x] = F[39,x]+F[47,x]
F[47,x] = 2*F[22,x]+F[48,x]+F[57,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[4,x]
F[53,x] = F[54,x]
F[54,x] = 2*F[22,x]+F[48,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[39,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[39,x]
F[59,x] = F[22,x]+F[60,x]+F[65,x]+F[67,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[51,x]
F[63,x] = F[64,x]
F[64,x] = 2*F[22,x]+F[55,x]+F[60,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[39,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[36,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[73,x]
F[71,x] = F[2,x]+F[72,x]
F[72,x] = F[40,x]
F[73,x] = F[36,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_{20}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{69}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = x^{2}
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{59}\! \left(x \right)
F_{36}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{57}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{39}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{39}\! \left(x \right)
F_{59}\! \left(x \right) = F_{22}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{51}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{55}\! \left(x \right)+F_{60}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{39}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{36}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{2}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{40}\! \left(x \right)
F_{73}\! \left(x \right) = F_{36}\! \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_20(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_15(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_69(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_35(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_34(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), x**2)
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_15(x)*F_4(x))
Eq(F_34(x), F_19(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_59(x))
Eq(F_36(x), F_22(x) + F_23(x) + F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_46(x))
Eq(F_42(x), F_4(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_4(x))
Eq(F_46(x), F_39(x) + F_47(x))
Eq(F_47(x), 2*F_22(x) + F_48(x) + F_57(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_4(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), 2*F_22(x) + F_48(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_39(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_39(x))
Eq(F_59(x), F_22(x) + F_60(x) + F_65(x) + F_67(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_51(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), 2*F_22(x) + F_55(x) + F_60(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_39(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_36(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_73(x))
Eq(F_71(x), F_2(x) + F_72(x))
Eq(F_72(x), F_40(x))
Eq(F_73(x), F_36(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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