0132_1203_1320_3021
Counting sequence:
1, 1, 2, 6, 20, 62, 178, 485, 1280, 3317, 8513, 21753, 55519, 141796, 362777, 930240, 2391182, 6161504, 15913240, 41185151, 106790414, 277351271, 721331447, 1878254691, 4895607085, 12770818102, 33337155743, 87072877290, 227529154940, 594776390402, 1555256512798, 4067771922953, 10641348370124, 27842314544201, 72856604233757, 190667368615485, 499020947561707, 1306137776032504, 3418859804591765, 8949342126115716, 23426898831023834, 61326681442538516, 160543524769849492, 420284101657710851, 1100268098273056250, 2880436630277747867, 7540870268746255283, 19741822332240130695, 51683875448346316825, 135308326269171089962, 354238077503167314683, 927399713790843220878, 2427948397495410369704, 6356419582997530509206, 16641257434201559555962, 43567244633216091268205, 114060255789064973095832, 298613072374016090971229, 781778042598659316238265, 2046719181924516871618881, 5358375684122407288439479, 14028400088222548897484236, 36726808727874551059869137, 96151993813598975290409736, 251729106996396612221084390, 659035193436696326978596232, 1725376201224217281498820048, 4517092856833633306231317575, 11825901244025294140912486310, 30960608587845983976521743247, 81055919870933151213845738543, 212207141580220503925725047403, 555565485685114523905395725509, 1454489276515599584115889118686, 3807902264762045640375714303767, 9969217357210076919443316529650, 26099749481024897799948715542932, 68330030424733308878652500184554, 178890340452022947701029544323990, 468340988211252440091520489701521, 1226132618666010320581786315186724, 3210056856604214690218518589829681, 8404037928479274632299472428564757, 22002056882894428061323990057033221, 57602132627116721441346051290162755, 150804340809863308402832012559295072, 394810889420452644768927167180574413, 1033628326677782101350586817170479180, 2706074089046125797062273873911662770, 7084593937288373939167447850135507132, 18547707716397182066647159500455655628, 48558529198904801848277537764790659403, 127127879854010997643371122952309936914, 332825110309896769392564479271476813891, 871347450967978527116494230945896701547, 2281217242376161365042691285183967431695, 5972304275719798914025984246741917374257, 15635695583891918548749787657114415254354, 40934782474153516035023865044348203523315, 107168651834924134080665647946627173604886, 280571173023250667093146495399333729914944
Generating function in Maple syntax:
(3*x^7-15*x^6+37*x^5-63*x^4+62*x^3-33*x^2+9*x-1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{3 x^{7}-15 x^{6}+37 x^{5}-63 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(x -1\right)^{3}}
Generating function in sympy syntax:
(3*x**7 - 15*x**6 + 37*x**5 - 63*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*(x-1)^3*F(x)-3*x^7+15*x^6-37*x^5+63*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(x -1\right)^{3} F \! \left(x \right)-3 x^{7}+15 x^{6}-37 x^{5}+63 x^{4}-62 x^{3}+33 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/4*2^n*n-1/2*n^2-3/4*2^n+1+1/2*n-1/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+1/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{2^{n} n}{4}-\frac{n^{2}}{2}-\frac{3 \,2^{n}}{4}+1+\frac{n}{2}-\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5} & \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) = 62
a(6) = 178
a(7) = 485
a(n+4) = 1/2*n^2-17*a(n+2)+7*a(n+3)-4*a(n)+16*a(n+1)-3/2*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) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 178
a \! \left(7\right) = 485
a \! \left(n +4\right) = \frac{n^{2}}{2}-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-4 a \! \left(n \right)+16 a \! \left(n +1\right)-\frac{3 n}{2}-3, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15567/
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[12,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[10,x]
F[19,x] = F[14,x]
F[20,x] = F[12,x]*F[13,x]
F[21,x] = F[22,x]+F[6,x]
F[22,x] = F[15,x]+F[23,x]+F[51,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[32,x]
F[25,x] = F[26,x]+F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[10,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[29,x]
F[32,x] = F[22,x]+F[33,x]
F[33,x] = 2*F[15,x]+F[34,x]+F[38,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[26,x]
F[37,x] = F[33,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[41,x]+F[45,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[18,x]+F[44,x]
F[44,x] = F[41,x]
F[45,x] = 2*F[15,x]+F[46,x]+F[50,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[30,x]
F[49,x] = F[45,x]
F[50,x] = F[12,x]*F[40,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[64,x]
F[53,x] = F[54,x]+F[6,x]
F[54,x] = F[15,x]+F[55,x]+F[63,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[10,x]+F[30,x]
F[58,x] = F[54,x]+F[59,x]
F[59,x] = 2*F[15,x]+F[50,x]+F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[48,x]+F[62,x]
F[62,x] = F[59,x]
F[63,x] = F[12,x]*F[53,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[42,x]
F[66,x] = 2*F[15,x]+F[46,x]+F[67,x]
F[67,x] = F[12,x]*F[64,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{51}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{26}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{30}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{64}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{6}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{63}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{48}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{59}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{42}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_21(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_10(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_12(x)*F_13(x))
Eq(F_21(x), F_22(x) + F_6(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_51(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_32(x))
Eq(F_25(x), F_26(x) + F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_10(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_29(x))
Eq(F_32(x), F_22(x) + F_33(x))
Eq(F_33(x), 2*F_15(x) + F_34(x) + F_38(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_26(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_41(x) + F_45(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_18(x) + F_44(x))
Eq(F_44(x), F_41(x))
Eq(F_45(x), 2*F_15(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_30(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_12(x)*F_40(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_64(x))
Eq(F_53(x), F_54(x) + F_6(x))
Eq(F_54(x), F_15(x) + F_55(x) + F_63(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_10(x) + F_30(x))
Eq(F_58(x), F_54(x) + F_59(x))
Eq(F_59(x), 2*F_15(x) + F_50(x) + F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_48(x) + F_62(x))
Eq(F_62(x), F_59(x))
Eq(F_63(x), F_12(x)*F_53(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_42(x))
Eq(F_66(x), 2*F_15(x) + F_46(x) + F_67(x))
Eq(F_67(x), F_12(x)*F_64(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, 3, 2], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 3, 2], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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