0123_0231_1230_1302_2301
Counting sequence:
1, 1, 2, 6, 19, 55, 145, 357, 840, 1920, 4308, 9548, 20977, 45773, 99307, 214359, 460574, 985422, 2100214, 4460326, 9442031, 19928931, 41950357, 88089241, 184560004, 385888620, 805321320, 1677739152, 3489681405, 7247781065, 15032412943, 31138544363, 64424545402, 133144027098, 274877953322, 566935735434, 1168231163419, 2405181751807, 4947802398809, 10170482639181, 20890721019136, 42880953584536, 87960930334012, 180319907078676, 369435907068489, 756464000058885, 1548112372071795, 3166593488173247, 6473924464539670, 13229323905612710, 27021597764453278, 55169095435538478, 112589990684533127, 229683580996188123, 468374361246847837, 954763121002886209, 1945555039024421564, 3963167672086431492, 8070450532248353104, 16429131440648024536, 33434723633599049941, 68022368771804493441, 138350580552822194967, 281312847124071257811, 571849066284996735474, 1162144876643702428850, 2361183241434823327570, 4796153459164484186642, 9739880870918644067635, 19774909647016640196855, 40140115104391985233313, 81460821829501380907893, 165282826900437583508152, 335288020283744811260848, 680020773533228911922852, 1378931012997936403614684, 2795640957858829967791009, 5666839779443574257788477, 11484795286338977161135099, 23271822027581611614596391, 47148106964970537815122446, 95505139749555704803451646, 193428131138340667954737222, 391691965555139852606638646, 793055337667196738609180959, 1605453488448227544011826515, 3249592603124123221612324645, 6576556458703582710403823337, 13307855422317837955167917300, 26925195854457020979058393500, 54469361728556732095564021048, 110176663496398844466024728608, 222829207071368449481845154477, 450610174299878420063284137273, 911123868914039882325758478367, 1842054778456645849049900028859, 3723723638170423866896568987754, 7526675438855112071386678746762, 15211807202738752817960442076794, 30740527055534562986295056494746, 62114879411183240673338460984651
Generating function in Maple syntax:
-(9*x^6-28*x^5+48*x^4-48*x^3+27*x^2-8*x+1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
-\frac{9 x^{6}-28 x^{5}+48 x^{4}-48 x^{3}+27 x^{2}-8 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-9*x**6 + 28*x**5 - 48*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)+9*x^6-28*x^5+48*x^4-48*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+9 x^{6}-28 x^{5}+48 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
2+11/24*n^2-1/4*n-1/4*n^3+1/24*n^4+2^(-1+n)*n-2^n
Explicit closed form in latex syntax:
2+\frac{11 n^{2}}{24}-\frac{n}{4}-\frac{n^{3}}{4}+\frac{n^{4}}{24}+2^{-1+n} n -2^{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) = 145
a(n+2) = 1/24*n^4-7/12*n^3+47/24*n^2-4*a(n)+4*a(n+1)-17/12*n+2, n >= 7
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) = 145
a \! \left(n +2\right) = \frac{n^{4}}{24}-\frac{7 n^{3}}{12}+\frac{47 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{17 n}{12}+2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17360/
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[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[14,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[10,x]+F[20,x]
F[20,x] = F[18,x]
F[21,x] = F[22,x]+F[36,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[9,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[15,x]+F[27,x]+F[31,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[10,x]
F[30,x] = F[26,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[10,x]+F[33,x]
F[33,x] = F[15,x]+F[31,x]+F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[10,x]+F[33,x]
F[36,x] = F[15,x]+F[37,x]+F[55,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[40,x]+F[6,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[10,x]+F[43,x]
F[43,x] = F[41,x]
F[44,x] = F[36,x]+F[45,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[40,x]
F[49,x] = F[45,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[33,x]+F[52,x]
F[52,x] = 2*F[15,x]+F[50,x]+F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[43,x]+F[52,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[32,x]+F[57,x]
F[57,x] = F[33,x]+F[58,x]
F[58,x] = F[15,x]+F[53,x]+F[59,x]+F[61,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[33,x]+F[58,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[20,x]+F[63,x]
F[63,x] = F[61,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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{36}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \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_{9}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \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_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{55}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{6}\! \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_{10}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)+F_{45}\! \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_{40}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{33}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{43}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{32}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{33}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{33}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{20}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{61}\! \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_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_14(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_10(x) + F_20(x))
Eq(F_20(x), F_18(x))
Eq(F_21(x), F_22(x) + F_36(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_9(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_15(x) + F_27(x) + F_31(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_10(x))
Eq(F_30(x), F_26(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_10(x) + F_33(x))
Eq(F_33(x), F_15(x) + F_31(x) + F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_10(x) + F_33(x))
Eq(F_36(x), F_15(x) + F_37(x) + F_55(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_40(x) + F_6(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_10(x) + F_43(x))
Eq(F_43(x), F_41(x))
Eq(F_44(x), F_36(x) + F_45(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_40(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_33(x) + F_52(x))
Eq(F_52(x), 2*F_15(x) + F_50(x) + F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_43(x) + F_52(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_32(x) + F_57(x))
Eq(F_57(x), F_33(x) + F_58(x))
Eq(F_58(x), F_15(x) + F_53(x) + F_59(x) + F_61(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_33(x) + F_58(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_20(x) + F_63(x))
Eq(F_63(x), F_61(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 2, 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, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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