0123_0132_0213_2103_2130
Counting sequence:
1, 1, 2, 6, 19, 55, 152, 403, 1043, 2647, 6627, 16425, 40414, 98913, 241159, 586331, 1422683, 3447023, 8343124, 20178563, 48777715, 117866099, 284734611, 687716601, 1660813394, 4010434329, 9683522255, 23380577743, 56449888291, 136289102983, 329042764112, 794399199571, 1917882261971, 4630232400943, 11178461710851, 26987347028265, 65153474372038, 157294826215857, 379744009237207, 916784311585763, 2213315069141483, 5343418494919295, 12900158769589516, 31143747160006787, 75187671525195571, 181519120741221899, 438225963542453427, 1057971131429487417, 2554168364645719226, 6166308089212414377, 14886784920556284575, 35939878553696004631, 86766543056948804659, 209472966365502694999, 505712478588545088776, 1220897928160311699091, 2947508342520378361235, 7115914625742116127367, 17179337614661887354659, 41474589889081353444777, 100128517448819490913198, 241731624878869990034049, 583591767358165475221159, 1408915159844557347772907, 3401422087457304783405179, 8211759335433210883913615, 19824940759431520283698372, 47861640856116477252999683, 115548222474654607574243443, 278958085810336552338005027, 673464394103391319571991507, 1625886874030356774691201785, 3925238142185831790036977954, 9476363158437673705559542713, 22877964459119674050087250799, 55232292076772973429016424671, 133342548612822985380944695363, 321917389302676981711180626343, 777177327218599990221184217024, 1876272043740570401940648838483, 4529721414700877275581185502035, 10935714873144187235491495487839, 26401151160992302871827753551171, 63738017195133791109318125388201, 153877185551268071429814172098454, 371492388297683340244902298132689, 896861962146656703425274770560375, 2165216312591032685573848213854323, 5227294587328780904144158124801291, 12619805487248690782195666858181663, 30466905561826320046311234805748092, 73553616610901588721082354565904131, 177574138783629919352541931378610611, 428701894178162117556677421977568443, 1034977927139955283320680992926651891, 2498657748458074530463722854692789305, 6032293424056107363509189416879028618, 14563244596570294194429830924606107977, 35158782617196703824139221341651827007, 84880809830963715038408934119190674983, 204920402279124155470972259491278264403
Generating function in Maple syntax:
-(x^9+x^8-2*x^7-3*x^6-3*x^5+3*x^4+6*x^3-3*x+1)/(x^2+2*x-1)/(x^2+x-1)^2
Generating function in latex syntax:
-\frac{x^{9}+x^{8}-2 x^{7}-3 x^{6}-3 x^{5}+3 x^{4}+6 x^{3}-3 x +1}{\left(x^{2}+2 x -1\right) \left(x^{2}+x -1\right)^{2}}
Generating function in sympy syntax:
(-x**9 - x**8 + 2*x**7 + 3*x**6 + 3*x**5 - 3*x**4 - 6*x**3 + 3*x - 1)/((x**2 + x - 1)**2*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+2*x-1)*(x^2+x-1)^2*F(x)+x^9+x^8-2*x^7-3*x^6-3*x^5+3*x^4+6*x^3-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+2 x -1\right) \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+x^{9}+x^{8}-2 x^{7}-3 x^{6}-3 x^{5}+3 x^{4}+6 x^{3}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,23,n = 1,-8,n = 2,3,n = 3,-1,0)+1/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)*n-1/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)*n+4/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-4/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-37/4*(-1-2^(1/2))^(-n)*2^(1/2)+37/4*(2^(1/2)-1)^(-n)*2^(1/2)+(-1/2*5^(1/2)-1/2)^(-n)+(1/2*5^(1/2)-1/2)^(-n)-12*(-1-2^(1/2))^(-n)-12*(2^(1/2)-1)^(-n)
Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}23 & n =0 \\ -8 & n =1 \\ 3 & n =2 \\ -1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}\, n}{5}-\frac{\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}\, n}{5}+\frac{4 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}-\frac{4 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}-\frac{37 \left(-1-\sqrt{2}\right)^{-n} \sqrt{2}}{4}+\frac{37 \left(\sqrt{2}-1\right)^{-n} \sqrt{2}}{4}+\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-12 \left(-1-\sqrt{2}\right)^{-n}-12 \left(\sqrt{2}-1\right)^{-n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 152
a(7) = 403
a(8) = 1043
a(9) = 2647
a(n+6) = a(n)+4*a(n+1)+2*a(n+2)-6*a(n+3)-2*a(n+4)+4*a(n+5), n >= 10
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) = 152
a \! \left(7\right) = 403
a \! \left(8\right) = 1043
a \! \left(9\right) = 2647
a \! \left(n +6\right) = a \! \left(n \right)+4 a \! \left(n +1\right)+2 a \! \left(n +2\right)-6 a \! \left(n +3\right)-2 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16028/
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[28,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[23,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[4,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[20,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[20,x]*F[4,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[37,x]
F[30,x] = F[2,x]+F[31,x]
F[31,x] = F[16,x]+F[32,x]+F[36,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]+F[4,x]
F[35,x] = F[13,x]+F[16,x]+F[21,x]
F[36,x] = F[2,x]*F[4,x]
F[37,x] = F[15,x]+F[38,x]
F[38,x] = 2*F[16,x]+F[39,x]+F[43,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]+F[42,x]
F[42,x] = F[27,x]
F[43,x] = F[15,x]*F[4,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_{28}\! \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_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \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_{13}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{27}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \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_28(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_23(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_20(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_20(x)*F_4(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_37(x))
Eq(F_30(x), F_2(x) + F_31(x))
Eq(F_31(x), F_16(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x) + F_4(x))
Eq(F_35(x), F_13(x) + F_16(x) + F_21(x))
Eq(F_36(x), F_2(x)*F_4(x))
Eq(F_37(x), F_15(x) + F_38(x))
Eq(F_38(x), 2*F_16(x) + F_39(x) + F_43(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x) + F_42(x))
Eq(F_42(x), F_27(x))
Eq(F_43(x), F_15(x)*F_4(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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": 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