0132_0231_1230_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 55, 148, 380, 949, 2333, 5686, 13794, 33383, 80691, 194920, 470712, 1136553, 2744057, 6624938, 15994238, 38613755, 93222127, 225058428, 543339444, 1311737821, 3166815637, 7645369694, 18457555674, 44560481743, 107578519915, 259717522384, 627013565552, 1513744654417, 3654502875377, 8822750406226, 21300003688950, 51422757785315, 124145519260839, 299713796308324, 723573111878892, 1746860020067589, 4217293152015629, 10181446324100486, 24580185800218322, 59341817924538935, 143263821649298083, 345869461223137080, 835002744095574312, 2015874949414287865, 4866752642924152297, 11749380235262594810, 28365513113449344366, 68480406462161286091, 165326326037771919199, 399133058537705127244, 963592443113182176548, 2326317944764069483309, 5616228332641321146245, 13558774610046711778990, 32733777552734744707530, 79026329715516201197471, 190786436983767147106011, 460599203683050495413152, 1111984844349868137936096, 2684568892382786771289249, 6481122629115441680518625, 15646814150613670132330658, 37774750930342781945184230, 91196316011299234022703539, 220167382952941249990595863, 531531081917181734003899956, 1283229546787304717998400604, 3097990175491791170000706133, 7479209897770887057999817981, 18056409971033565286000347350, 43592029839838017630000518082, 105240469650709600546001389063, 254072969141257218722003301907, 613386407933224037990007998728, 1480845785007705294702019305368, 3575077977948634627394046615625, 8631001740904974549490112542937, 20837081459758583726374271707978, 50305164660422142002238655965534, 121447410780602867730851583645851, 293199986221627877463941823264207, 707847383223858622658735230181404, 1708894752669345122781412283634324, 4125636888562548868221559797457533, 9960168529794442859224531878557045, 24045973948151434586670623554579454, 58052116426097312032565778987723962, 140150206800346058651802181530035567, 338352530026789429336170142047803467, 816855266853924917324142465625651056, 1972063063734639263984455073299114320, 4760981394323203445293052612223888625, 11494025852381046154570560297746900689, 27749033099085295754434173207717699314, 66992092050551637663438906713182308822, 161733217200188571081311986634082326659
Generating function in Maple syntax:
(5*x^2-4*x+1)/(x^2+2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{5 x^{2}-4 x +1}{\left(x^{2}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(5*x**2 - 4*x + 1)/((x - 1)**3*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+2*x-1)*(x-1)^3*F(x)-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
1/4*(2-2^(1/2))*(-1-2^(1/2))^(-n)+1/4*(2+2^(1/2))*(2^(1/2)-1)^(-n)-1/2*n^2-1/2*n
Explicit closed form in latex syntax:
\frac{\left(2-\sqrt{2}\right) \left(-1-\sqrt{2}\right)^{-n}}{4}+\frac{\left(2+\sqrt{2}\right) \left(\sqrt{2}-1\right)^{-n}}{4}-\frac{n^{2}}{2}-\frac{n}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+2) = n^2+2*a(n+1)+a(n)+n-1, n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(n +2\right) = n^{2}+2 a \! \left(n +1\right)+a \! \left(n \right)+n -1, \quad n \geq 3
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19064/
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[22,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[30,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[23,x]
F[30,x] = F[18,x]+F[31,x]+F[48,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[37,x]
F[33,x] = F[34,x]+F[6,x]
F[34,x] = F[18,x]+F[19,x]+F[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[13,x]
F[37,x] = F[30,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[18,x]+F[42,x]+F[46,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[20,x]+F[44,x]
F[44,x] = F[41,x]+F[45,x]
F[45,x] = F[39,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[23,x]+F[41,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[47,x]+F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[14,x]
F[54,x] = 2*F[18,x]+F[55,x]+F[57,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[17,x]
F[57,x] = F[12,x]*F[50,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_{22}\! \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_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{48}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{6}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right)
F_{37}\! \left(x \right) = F_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{20}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{23}\! \left(x \right)+F_{41}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{47}\! \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_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{14}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{17}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \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_22(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_30(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_23(x))
Eq(F_30(x), F_18(x) + F_31(x) + F_48(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_37(x))
Eq(F_33(x), F_34(x) + F_6(x))
Eq(F_34(x), F_18(x) + F_19(x) + F_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_13(x))
Eq(F_37(x), F_30(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_18(x) + F_42(x) + F_46(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_20(x) + F_44(x))
Eq(F_44(x), F_41(x) + F_45(x))
Eq(F_45(x), F_39(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_23(x) + F_41(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_47(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_12(x)*F_53(x))
Eq(F_53(x), F_14(x))
Eq(F_54(x), 2*F_18(x) + F_55(x) + F_57(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_17(x))
Eq(F_57(x), F_12(x)*F_50(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": [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": [2, 1, 3, 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": [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": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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