0123_0132_0213_1032_1230
Counting sequence:
1, 1, 2, 6, 19, 57, 163, 453, 1236, 3332, 8909, 23685, 62713, 165567, 436180, 1147302, 3014295, 7912577, 20757191, 54426129, 142654852, 373804752, 979290457, 2565126141, 6718202829, 17593705307, 46071348068, 120637191138, 315873901171, 827051817657, 2165416087339, 5669465394045, 14843517797748, 38862163087292, 101745121126085, 266377498782837, 697395970859905, 1825827602893527, 4780121212489396, 12514604778209214, 32763830600179887, 85777161963483377, 224568205148418287, 587928553158979401, 1539219653619689764, 4029734806180095432, 10549993561715027473, 27620263472285934573, 72310832041351176981, 189312303023482989107, 495626217771393672644, 1297566631773453481530, 3397074240511506462283, 8893657215681337759017, 23283899658365271780115, 60958046263067421242997, 159590248138122514261140, 417812716165838214884852, 1093847936388415000791101, 2863731165057366260712549, 7497345702899463143932297, 19628306231872356044822127, 51387573569179905302713492, 134534415628591369200947286, 352215675622441264253360199, 922112615850424999456841057, 2414122181152216381181796503, 6320253946052985385478277633, 16546639693900255700562174916, 43319665209434802956616666432, 113412356081978178482424485449, 296917403331647755339040076765, 777339854503261088285892105693, 2035102161358727438297568713867, 5327966631934104966495648522020, 13948797739165954750574555115282, 36518426595008493555936852179875, 95606482064748993960199660257177, 250301019637017423604073913207931, 655296576921861146105876103670509, 1715588711279681751109777368433332, 4491469557219415576599447319982252, 11758819960983027911912607527334197, 30784990326938594016642066809870421, 80596151022250605838548641973301009, 211003462744648926876657228903612807, 552414237221367581508840721951553908, 1446239248938796631023394500657470318, 3786303509633707938209207054338944351, 9912671280039698436739454567179170257, 25951710330640129878019916959927058879, 67942459712190176208921620797331501625, 177875668806549368771267702763887824228, 465684546708695870148826914379606625592, 1219177971322014121761323914729476647425, 3191849367262298255304486511323541690317, 8356370130474784164486158935660209640485, 21877261024181861278814496814690463318627, 57275412942110413753266144386062810613636, 149948977802228608143581821824581360895594, 392571520464733867002643151615213066719867
Generating function in Maple syntax:
-(2*x^6+2*x^5+4*x^4-x^3-7*x^2+5*x-1)/(2*x-1)/(x^2-3*x+1)/(x^2+x-1)
Generating function in latex syntax:
-\frac{2 x^{6}+2 x^{5}+4 x^{4}-x^{3}-7 x^{2}+5 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right)}
Generating function in sympy syntax:
(-2*x**6 - 2*x**5 - 4*x**4 + x**3 + 7*x**2 - 5*x + 1)/((2*x - 1)*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x^2+x-1)*F(x)+2*x^6+2*x^5+4*x^4-x^3-7*x^2+5*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^{2}+x -1\right) F \! \left(x \right)+2 x^{6}+2 x^{5}+4 x^{4}-x^{3}-7 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(-19*5^(1/2)+55)*(3/2-1/2*5^(1/2))^(-n)+1/20*(19*5^(1/2)+55)*(3/2+1/2*5^(1/2))^(-n)+1/20*(5^(1/2)-5)*(-1/2*5^(1/2)-1/2)^(-n)+1/20*(-5^(1/2)-5)*(1/2*5^(1/2)-1/2)^(-n)-2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-19 \sqrt{5}+55\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(19 \sqrt{5}+55\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(\sqrt{5}-5\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\frac{\left(-\sqrt{5}-5\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}-2^{-1+n} & \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) = 57
a(6) = 163
a(n+5) = -2*a(n)+5*a(n+1)+4*a(n+2)-11*a(n+3)+6*a(n+4), 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) = 57
a \! \left(6\right) = 163
a \! \left(n +5\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)+4 a \! \left(n +2\right)-11 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15977/
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[14,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[4,x]*F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[62,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[29,x]
F[19,x] = F[20,x]+F[22,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[4,x]
F[22,x] = F[16,x]+F[23,x]+F[27,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[4,x]
F[26,x] = F[12,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[7,x]
F[29,x] = F[30,x]+F[52,x]
F[30,x] = F[16,x]+F[17,x]+F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[2,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[40,x]
F[36,x] = F[37,x]+F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[25,x]
F[40,x] = F[33,x]+F[41,x]
F[41,x] = 2*F[16,x]+F[42,x]+F[50,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[46,x]
F[44,x] = F[45,x]
F[45,x] = F[25,x]*F[4,x]
F[46,x] = F[47,x]
F[47,x] = 2*F[16,x]+F[42,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[33,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[49,x]
F[52,x] = F[16,x]+F[53,x]+F[58,x]+F[60,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[45,x]
F[56,x] = F[57,x]
F[57,x] = 2*F[16,x]+F[48,x]+F[53,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[49,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[15,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[32,x]+F[64,x]
F[64,x] = F[15,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_{14}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{62}\! \left(x \right)
F_{16}\! \left(x \right) = 0
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_{29}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
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_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{52}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{25}\! \left(x \right)
F_{40}\! \left(x \right) = F_{33}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{42}\! \left(x \right)+F_{50}\! \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_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{33}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{16}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{45}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{49}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{32}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \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_14(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_4(x)*F_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_62(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_29(x))
Eq(F_19(x), F_20(x) + F_22(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_4(x))
Eq(F_22(x), F_16(x) + F_23(x) + F_27(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_4(x))
Eq(F_26(x), F_12(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_7(x))
Eq(F_29(x), F_30(x) + F_52(x))
Eq(F_30(x), F_16(x) + F_17(x) + F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_2(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_40(x))
Eq(F_36(x), F_37(x) + F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_25(x))
Eq(F_40(x), F_33(x) + F_41(x))
Eq(F_41(x), 2*F_16(x) + F_42(x) + F_50(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_46(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_25(x)*F_4(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), 2*F_16(x) + F_42(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_33(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_49(x))
Eq(F_52(x), F_16(x) + F_53(x) + F_58(x) + F_60(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_45(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), 2*F_16(x) + F_48(x) + F_53(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_49(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_15(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_32(x) + F_64(x))
Eq(F_64(x), F_15(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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