0231_0312_1230_2103_3012
Counting sequence:
1, 1, 2, 6, 19, 50, 120, 289, 706, 1758, 4447, 11382, 29368, 76181, 198294, 517270, 1351195, 3532554, 9240376, 24178713, 63279802, 165634862, 433582983, 1135046446, 2971446904, 7779117165, 20365618030, 53317273254, 139585451491, 365437867298, 956726186232, 2504737513297, 6557481211378, 17167697800446, 44945598727279, 117669076598310, 308061595821880, 806515653838469, 2111485273418886, 5527940017114678, 14472334536346987, 37889063201044602, 99194854434326968, 259695499078594761, 679891641145655914, 1779979421679230030, 4660046619557089815, 12200160429977952094, 31940434659027734776, 83621143528742133213, 218922995897486514142, 573147844115642139462, 1500520536371652483763, 3928413764873452621586, 10284720758045055270264, 26925748508932200388225, 70492524768218382982690, 184551825794860272847134, 483162952614966596934271, 1264937032047781003618518, 3311648143524722060959672, 8670007398520472311961717, 22698374052027127654665078, 59425114757545430564474326, 155576970220584116730938299, 407305795904166392232961770, 1066340417491849485264748600, 2791715456571275961462706809, 7308805952221806722321596186, 19134702400093866426601728878, 50095301248059343101781461927, 131151201344083435644140175502, 343358302784189787140334454648, 898923707008484021851956097101, 2353412818241259197800322135374, 6161314747715288587008891516390, 16130531424904598498071021919875, 42230279526998493857508724956674, 110560307156090861959604373169656, 289450641941274057856758165485233, 757791618667731256331273114438482, 1983924214061919621693117939915582, 5193981023518027464024740458546063, 13598018856492162536213819951045766, 35600075545958459765726095663152184, 93202207781383216147906559822294885, 244006547798191187686045052856177510, 638817435613190345305222160582566774, 1672445759041379845632666459780296971, 4378519841510949187390815811483427418, 11463113765491467709740864598283862712, 30010821454963453930830900199707141417, 78569350599398894064952041840790419658, 205697230343233228235224553378955956366, 538522340430300790594121152192322146359, 1409869790947669143471737765150547018430, 3691087032412706639699090539108100141560, 9663391306290450775428131109975071174589, 25299086886458645686265898444467212383166, 66233869353085486282852757134877982744198, 173402521172797813161456161525268251619667
Generating function in Maple syntax:
(4*x^7-8*x^6-7*x^5-x^4+2*x^3+4*x^2-4*x+1)/(x-1)/(x^2-3*x+1)/(x^2+x-1)
Generating function in latex syntax:
\frac{4 x^{7}-8 x^{6}-7 x^{5}-x^{4}+2 x^{3}+4 x^{2}-4 x +1}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right)}
Generating function in sympy syntax:
(4*x**7 - 8*x**6 - 7*x**5 - x**4 + 2*x**3 + 4*x**2 - 4*x + 1)/((x - 1)*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^2-3*x+1)*(x^2+x-1)*F(x)+4*x^7-8*x^6-7*x^5-x^4+2*x^3+4*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+4 x^{7}-8 x^{6}-7 x^{5}-x^{4}+2 x^{3}+4 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,-35/2*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)+92/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+35/2*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)-92/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-174/5*(3/2-1/2*5^(1/2))^(1-n)-174/5*(1/2*5^(1/2)-1/2)^(1-n)-174/5*(3/2+1/2*5^(1/2))^(1-n)-174/5*(-1/2*5^(1/2)-1/2)^(1-n)+527/10*(3/2-1/2*5^(1/2))^(-n)-87/5*(1/2*5^(1/2)-1/2)^(-n)+527/10*(3/2+1/2*5^(1/2))^(-n)-87/5*(-1/2*5^(1/2)-1/2)^(-n)-9)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ -\frac{35 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{2}+\frac{92 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\frac{35 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{2}-\\\frac{92 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}-\frac{174 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}}{5}-\frac{174 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}}{5}-\\\frac{174 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}}{5}-\frac{174 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}}{5}+\frac{527 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}-\\\frac{87 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+\frac{527 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}-\frac{87 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}-9 & \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) = 50
a(6) = 120
a(7) = 289
a(n+4) = a(n)-2*a(n+1)-3*a(n+2)+4*a(n+3)-9, 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) = 50
a \! \left(6\right) = 120
a \! \left(7\right) = 289
a \! \left(n +4\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right)-9, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20725/
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[17,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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[38,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[37,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[30,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[11,x]+F[27,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[18,x]+F[31,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[22,x]+F[34,x]
F[37,x] = F[21,x]*F[4,x]
F[38,x] = F[23,x]+F[39,x]+F[57,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[46,x]
F[41,x] = F[42,x]+F[7,x]
F[42,x] = F[15,x]+F[23,x]+F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[27,x]
F[46,x] = F[47,x]+F[52,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[18,x]
F[51,x] = F[31,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[22,x]
F[56,x] = F[34,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[64,x]
F[59,x] = F[18,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[14,x]+F[7,x]
F[64,x] = F[22,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[14,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_{17}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \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_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{22}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{23}\! \left(x \right)+F_{39}\! \left(x \right)+F_{57}\! \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_{46}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{27}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{52}\! \left(x \right)
F_{47}\! \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_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)
F_{51}\! \left(x \right) = F_{31}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \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_{22}\! \left(x \right)
F_{56}\! \left(x \right) = F_{34}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{64}\! \left(x \right)
F_{59}\! \left(x \right) = F_{18}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{64}\! \left(x \right) = F_{22}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{14}\! \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_17(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_6(x))
Eq(F_10(x), F_11(x) + F_14(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_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_38(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_37(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_30(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_11(x) + F_27(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_18(x) + F_31(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_22(x) + F_34(x))
Eq(F_37(x), F_21(x)*F_4(x))
Eq(F_38(x), F_23(x) + F_39(x) + F_57(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_46(x))
Eq(F_41(x), F_42(x) + F_7(x))
Eq(F_42(x), F_15(x) + F_23(x) + F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_27(x))
Eq(F_46(x), F_47(x) + F_52(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_18(x))
Eq(F_51(x), F_31(x))
Eq(F_52(x), F_53(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_22(x))
Eq(F_56(x), F_34(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_64(x))
Eq(F_59(x), F_18(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_14(x) + F_7(x))
Eq(F_64(x), F_22(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_14(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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