0213_0231_1203_1230_3120
Counting sequence:
1, 1, 2, 6, 19, 60, 186, 562, 1654, 4755, 13406, 37209, 102005, 276931, 746117, 1998192, 5326177, 14144058, 37449924, 98926456, 260836600, 686728881, 1805903432, 4744612671, 12456245879, 32682844945, 85713923471, 224710583382, 588937920019, 1543176918120, 4042807426446, 10589808763534, 27736014104650, 72637560902751, 190216397050562, 498093234626997, 1304230810554377, 3414942794419183, 8941301947339049, 23410406156608716, 61292871459986005, 160474255537301286, 420142264657729512, 1099977827203325236, 2879842893998750764, 7539656407909192605, 19739341834007870156, 51678808898765527227, 135297982063776975659, 354216966879914017885, 927356648119406493011, 2427860577302676647634, 6356240564912342534695, 16640892642631742548596, 43566501539277575139282, 114058742579590176612202, 298609991911870969555102, 781771773587978016511083, 2046706427730372158377574, 5358349743388553634386001, 14028347342063713134238685, 36726701506174622623104091, 96151775911434604596335117, 251728664274538843191849576, 659034294157922733637959049, 1725374374996553984253214290, 4517089149038074490611447116, 11825893717753712067415431472, 30960593313941890945013009824, 81055888880403107381799028425, 212207078713716700723573144976, 555565358181219486424974960759, 1454489017966034647002813674015, 3807901740579366041845095588841, 9969216294677618273773143453527, 26099747327625781611390498103182, 68330026061266678707100322735227, 178890331611752891769053704291008, 468340970304038737049825839372374, 1226132582398235732142911074007686, 3210056783161971148629796226435890, 8404037779781398819700083939711671, 22002056581871898977281325555202122, 57602132017818108355572947238256941, 150804339576758972395910254359002609, 394810886925229753084332550587034375, 1033628321629307878639895383597495601, 2706074078833120472957886605993895492, 7084593916630249533592664512756379389, 18547707674616705740765575222610220990, 48558529114415394167050334002925439216, 127127879683175272221988645016230802284, 332825109964511498431943382574621163860, 871347450269780344959539755902790647861, 2281217240964909720257357771798965825172, 5972304272867585063512468093374335171187, 15635695578128069725837057097184092876627, 40934782462506976145427007418097242833333, 107168651811393369813929139681344620364699, 280571172975713769584587892843207359723146, 734544867123908439678803765620412485684495
Generating function in Maple syntax:
-(x^7+2*x^6-29*x^5+62*x^4-62*x^3+33*x^2-9*x+1)/(x^2-3*x+1)/(2*x-1)^2/(-1+x)^3
Generating function in latex syntax:
-\frac{x^{7}+2 x^{6}-29 x^{5}+62 x^{4}-62 x^{3}+33 x^{2}-9 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(-x**7 - 2*x**6 + 29*x**5 - 62*x**4 + 62*x**3 - 33*x**2 + 9*x - 1)/((x - 1)**3*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*(-1+x)^3*F(x)+x^7+2*x^6-29*x^5+62*x^4-62*x^3+33*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(-1+x \right)^{3} F \! \left(x \right)+x^{7}+2 x^{6}-29 x^{5}+62 x^{4}-62 x^{3}+33 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/20*(6*5^(1/2)+10)*(3/2-1/2*5^(1/2))^(-n)+1/20*(-6*5^(1/2)+10)*(3/2+1/2*5^(1/2))^(-n)+1/20*(-5*n-35)*2^n+1/2*n^2-1/2*n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(6 \sqrt{5}+10\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(-6 \sqrt{5}+10\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(-5 n -35\right) 2^{n}}{20}+\frac{n^{2}}{2}-\frac{n}{2}+2 & \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) = 60
a(6) = 186
a(7) = 562
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-1/2*n*(n-3), 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) = 60
a \! \left(6\right) = 186
a \! \left(7\right) = 562
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{n \left(n -3\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: row_placements
Tree: http://www.permpal.com/tree/21871/
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[53,x]+F[54,x]
F[37,x] = F[38,x]*F[9,x]
F[38,x] = F[1,x]+F[37,x]+F[39,x]+F[41,x]
F[39,x] = F[40,x]*F[9,x]
F[40,x] = F[1,x]+F[21,x]+F[39,x]
F[41,x] = F[42,x]
F[42,x] = F[16,x]*F[43,x]*F[9,x]
F[43,x] = F[23,x]+F[44,x]
F[44,x] = F[27,x]+F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]*F[9,x]
F[48,x] = F[14,x]+F[46,x]
F[49,x] = 2*F[29,x]+F[50,x]+F[52,x]
F[50,x] = F[51,x]*F[9,x]
F[51,x] = F[28,x]+F[49,x]
F[52,x] = F[45,x]*F[9,x]
F[53,x] = F[36,x]*F[9,x]
F[54,x] = F[55,x]*F[9,x]
F[55,x] = 2*F[56,x]+F[1,x]+F[4,x]+F[6,x]
F[56,x] = F[55,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_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{9}\! \left(x \right)
F_{38}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{9}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)+F_{39}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{16}\! \left(x \right) F_{43}\! \left(x \right) F_{9}\! \left(x \right)
F_{43}\! \left(x \right) = F_{23}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{27}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right) F_{9}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right) F_{9}\! \left(x \right)
F_{51}\! \left(x \right) = F_{28}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{45}\! \left(x \right) F_{9}\! \left(x \right)
F_{53}\! \left(x \right) = F_{36}\! \left(x \right) F_{9}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right) F_{9}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{56}\! \left(x \right)+F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{6}\! \left(x \right)
F_{56}\! \left(x \right) = F_{55}\! \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_53(x) + F_54(x))
Eq(F_37(x), F_38(x)*F_9(x))
Eq(F_38(x), F_1(x) + F_37(x) + F_39(x) + F_41(x))
Eq(F_39(x), F_40(x)*F_9(x))
Eq(F_40(x), F_1(x) + F_21(x) + F_39(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_16(x)*F_43(x)*F_9(x))
Eq(F_43(x), F_23(x) + F_44(x))
Eq(F_44(x), F_27(x) + F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x)*F_9(x))
Eq(F_48(x), F_14(x) + F_46(x))
Eq(F_49(x), 2*F_29(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_51(x)*F_9(x))
Eq(F_51(x), F_28(x) + F_49(x))
Eq(F_52(x), F_45(x)*F_9(x))
Eq(F_53(x), F_36(x)*F_9(x))
Eq(F_54(x), F_55(x)*F_9(x))
Eq(F_55(x), F_1(x) + F_4(x) + 2*F_56(x) + F_6(x))
Eq(F_56(x), F_55(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], 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