0213_0231_1203_3120
Counting sequence:
1, 1, 2, 6, 20, 65, 203, 612, 1791, 5114, 14315, 39445, 107370, 289532, 775182, 2064214, 5474226, 14472453, 38171673, 100500328, 264245653, 694070286, 1821633657, 4778168921, 12527551120, 32833842240, 86032693228, 225381675002, 590347209496, 1546129711849, 4048981446055, 10602693669980, 27762857655275, 72693395483122, 190332361173607, 498333752802189, 1304729026767926, 3415973586578068, 8943432251126378, 23414804203129038, 61301942430925118, 160492947234984221, 420180747564713237, 1100056992040537536, 2880005621719674945, 7539990659444050790, 19740027929263617653, 51680216273649096817, 135300867182288277276, 354222877854424960184, 927368751543405070920, 2427885347100627206290, 6356291230408150474980, 16640996225423172093537, 43566713208460061577507, 114059174925154404206164, 298610874617397934200151, 781773575027828964738858, 2046710102667668092733283, 5358357237378333578923973, 14028362618273649174995458, 36726732635055247008008556, 96151839322117357972956742, 251728793401747359158750726, 659034557024025783999111370, 1725374909952132121830254965, 4517090237395974839475038353, 11825895931363000912561673432, 30960597814947444930143654381, 81055898029988167941736682494, 212207097308034727022801228785, 555565395960151349382136727625, 1454489094704489993634548456360, 3807901896417459976543387703152, 9969216611076172626039372836132, 26099747969867623281662247233514, 68330027364639827979122401785776, 178890334256278122176055024034073, 468340975668647061589742802207263, 1226132593278568108674573646442508, 3210056805224867356596778664905923, 8404037824511654145441363403925666, 22002056672541335448378513658254047, 57602132201574832936996581793687741, 150804339949108124837216040168600270, 394810887679599464523861155603787140, 1033628323157390114632786660426204578, 2706074081927970571171337293241812222, 7084593922897320982474902154433305686, 18547707687305591143440723040326358029, 48558529140102649982221974707082384525, 127127879735168753871974616561994141360, 332825110069736401771200705941046848665, 871347450482706031716625163185440144342, 2281217241395712853928670107463861189357, 5972304273739094851169375806903318763489, 15635695579890896341779438608640445914676, 40934782466072243458568902609806720746088, 107168651818603132602727194402357119997568, 280571172990291751487212530960419446742018, 734544867153381316134110099205210835371388
Generating function in Maple syntax:
(x^7-6*x^6+19*x^5-39*x^4+43*x^3-26*x^2+8*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{7}-6 x^{6}+19 x^{5}-39 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**7 - 6*x**6 + 19*x**5 - 39*x**4 + 43*x**3 - 26*x**2 + 8*x - 1)/((x - 1)**4*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^4*F(x)-x^7+6*x^6-19*x^5+39*x^4-43*x^3+26*x^2-8*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 -1\right)^{4} F \! \left(x \right)-x^{7}+6 x^{6}-19 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(9*5^(1/2)+15)*(3/2-1/2*5^(1/2))^(-n)+1/30*(-9*5^(1/2)+15)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3+11/6*n-7*2^(n-1)+3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(9 \sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-9 \sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}+\frac{11 n}{6}-\\7 \,2^{n -1}+3 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 203
a(7) = 612
a(n+3) = 1/6*n^3+2*a(n)-7*a(n+1)+5*a(n+2)-1/6*n+2, 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) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 203
a \! \left(7\right) = 612
a \! \left(n +3\right) = \frac{n^{3}}{6}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{n}{6}+2, \quad n \geq 8
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21330/
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[9,x]
F[3,x] = F[1,x]+F[35,x]+F[4,x]
F[4,x] = F[5,x]*F[9,x]
F[5,x] = 2*F[4,x]+F[1,x]+F[6,x]
F[6,x] = F[7,x]*F[9,x]
F[7,x] = F[1,x]+F[33,x]+F[34,x]+F[8,x]
F[8,x] = F[10,x]*F[9,x]
F[9,x] = x
F[10,x] = F[1,x]+F[11,x]+F[21,x]+F[8,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[16,x]*F[9,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[9,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[9,x]
F[19,x] = F[16,x]+F[20,x]
F[20,x] = F[17,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]*F[9,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[9,x]
F[27,x] = F[14,x]+F[28,x]
F[28,x] = F[29,x]+F[30,x]+F[32,x]
F[29,x] = 0
F[30,x] = F[31,x]*F[9,x]
F[31,x] = F[25,x]+F[28,x]
F[32,x] = F[27,x]*F[9,x]
F[33,x] = F[7,x]*F[9,x]
F[34,x] = F[5,x]*F[9,x]
F[35,x] = F[36,x]*F[9,x]
F[36,x] = F[1,x]+F[37,x]+F[60,x]+F[61,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[9,x]
F[39,x] = F[40,x]+F[58,x]
F[40,x] = F[1,x]+F[41,x]+F[42,x]+F[46,x]
F[41,x] = F[40,x]*F[9,x]
F[42,x] = F[43,x]*F[9,x]
F[43,x] = F[1,x]+F[44,x]+F[45,x]
F[44,x] = F[43,x]*F[9,x]
F[45,x] = F[10,x]*F[9,x]
F[46,x] = F[47,x]
F[47,x] = F[16,x]*F[48,x]*F[9,x]
F[48,x] = F[23,x]+F[49,x]
F[49,x] = F[27,x]+F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]*F[9,x]
F[53,x] = F[14,x]+F[51,x]
F[54,x] = 2*F[29,x]+F[55,x]+F[57,x]
F[55,x] = F[56,x]*F[9,x]
F[56,x] = F[28,x]+F[54,x]
F[57,x] = F[50,x]*F[9,x]
F[58,x] = F[59,x]
F[59,x] = F[13,x]*F[24,x]*F[43,x]*F[9,x]
F[60,x] = F[36,x]*F[9,x]
F[61,x] = F[62,x]*F[9,x]
F[62,x] = 2*F[63,x]+F[1,x]+F[4,x]+F[6,x]
F[63,x] = F[62,x]*F[9,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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{35}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{5}\! \left(x \right) = 2 F_{4}\! \left(x \right)+F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right) F_{9}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{10}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{21}\! \left(x \right)+F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{9}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{9}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right) F_{9}\! \left(x \right)
F_{33}\! \left(x \right) = F_{7}\! \left(x \right) F_{9}\! \left(x \right)
F_{34}\! \left(x \right) = F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{9}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{9}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{58}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{41}\! \left(x \right) = F_{40}\! \left(x \right) F_{9}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right) F_{9}\! \left(x \right)
F_{43}\! \left(x \right) = F_{1}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{43}\! \left(x \right) F_{9}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right) F_{9}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right) F_{48}\! \left(x \right) F_{9}\! \left(x \right)
F_{48}\! \left(x \right) = F_{23}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{27}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right) F_{9}\! \left(x \right)
F_{53}\! \left(x \right) = F_{14}\! \left(x \right)+F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right) F_{9}\! \left(x \right)
F_{56}\! \left(x \right) = F_{28}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{50}\! \left(x \right) F_{9}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right) F_{43}\! \left(x \right) F_{9}\! \left(x \right)
F_{60}\! \left(x \right) = F_{36}\! \left(x \right) F_{9}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right) F_{9}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{63}\! \left(x \right)+F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{6}\! \left(x \right)
F_{63}\! \left(x \right) = F_{62}\! \left(x \right) F_{9}\! \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)*F_9(x))
Eq(F_3(x), F_1(x) + F_35(x) + F_4(x))
Eq(F_4(x), F_5(x)*F_9(x))
Eq(F_5(x), F_1(x) + 2*F_4(x) + F_6(x))
Eq(F_6(x), F_7(x)*F_9(x))
Eq(F_7(x), F_1(x) + F_33(x) + F_34(x) + F_8(x))
Eq(F_8(x), F_10(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_1(x) + F_11(x) + F_21(x) + F_8(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_16(x)*F_9(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_9(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_9(x))
Eq(F_19(x), F_16(x) + F_20(x))
Eq(F_20(x), F_17(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x)*F_9(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_9(x))
Eq(F_27(x), F_14(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_32(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_31(x)*F_9(x))
Eq(F_31(x), F_25(x) + F_28(x))
Eq(F_32(x), F_27(x)*F_9(x))
Eq(F_33(x), F_7(x)*F_9(x))
Eq(F_34(x), F_5(x)*F_9(x))
Eq(F_35(x), F_36(x)*F_9(x))
Eq(F_36(x), F_1(x) + F_37(x) + F_60(x) + F_61(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_9(x))
Eq(F_39(x), F_40(x) + F_58(x))
Eq(F_40(x), F_1(x) + F_41(x) + F_42(x) + F_46(x))
Eq(F_41(x), F_40(x)*F_9(x))
Eq(F_42(x), F_43(x)*F_9(x))
Eq(F_43(x), F_1(x) + F_44(x) + F_45(x))
Eq(F_44(x), F_43(x)*F_9(x))
Eq(F_45(x), F_10(x)*F_9(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_16(x)*F_48(x)*F_9(x))
Eq(F_48(x), F_23(x) + F_49(x))
Eq(F_49(x), F_27(x) + F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x)*F_9(x))
Eq(F_53(x), F_14(x) + F_51(x))
Eq(F_54(x), 2*F_29(x) + F_55(x) + F_57(x))
Eq(F_55(x), F_56(x)*F_9(x))
Eq(F_56(x), F_28(x) + F_54(x))
Eq(F_57(x), F_50(x)*F_9(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_13(x)*F_24(x)*F_43(x)*F_9(x))
Eq(F_60(x), F_36(x)*F_9(x))
Eq(F_61(x), F_62(x)*F_9(x))
Eq(F_62(x), F_1(x) + F_4(x) + F_6(x) + 2*F_63(x))
Eq(F_63(x), F_62(x)*F_9(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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