0132_0321_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 62, 176, 469, 1201, 3005, 7428, 18270, 44920, 110721, 274073, 681969, 1706492, 4294370, 10865576, 27630901, 70585657, 181046237, 465999852, 1203060126, 3113820880, 8076537057, 20985767441, 54608136129, 142268216276, 371005331090, 968264749808, 2528621536645, 6606862442785, 17269685649821, 45156023609268, 118102822156734, 308954874366472, 808353779050785, 2115264649197161, 5535705001754481, 14488276943277452, 37921772844068642, 99261923304048152, 259832935864592821, 680173112615321065, 1780555560097666877, 4661225286840504156, 12202570544609959614, 31945360447082534176, 83631206220265826625, 218943543507855044513, 573189783785346969345, 1500606104599816120100, 3928588279092431769746, 10285076541984279680288, 26926473587774287722949, 70494001947766903333777, 184554834197584179854525, 483169077507503387156196, 1264949498006765955017886, 3311673507790083367810072, 8670058991745279814450497, 22698478967863783180046777, 59425328047990995479561841, 155577403719016797856108700, 407306676736110073463159522, 1066342206825885745641074696, 2791719090579633754393306549, 7308813330919072588840997401, 19134717378849465933605331485, 50095331648292624612668974668, 131151263029994077821274923870, 343358427926899090915873216240, 898923960835678443517721081121, 2353413332979199305743925779633, 6161315791358270743775093847489, 16130533540524765741398963116532, 42230283814907232663448074563090, 110560315845245145717779947407056, 289450659546256233633125398756741, 757791654331042817879192025926017, 1983924286295237154221899262883677, 5193981169798051330865916989928852, 13598019152678987845823702280274686, 35600076145585665492078947235573160, 93202208995144737742516766671675617, 244006550254728451041990832221671561, 638817440584293312161119138781293809, 1672445769099642659325942301505164076, 4378519861859588574244357000114203170, 11463113806652974001579422766825654648, 30010821538214921549472490634795585845, 78569350767758739370063190203339573705, 205697230683666738977703326900605725821, 538522341118595452338091213936894090428, 1409869792339113747468803020962212137854, 3691087035225306408697070979404618754112, 9663391311975071435408491614862607487105, 25299086897946729250157718133807551442241, 66233869376299337898437593070613584337921, 173402521219700885368129525139749960633412
Generating function in Maple syntax:
(x^8-x^7-13*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{x^{8}-x^{7}-13 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:
(x**8 - x**7 - 13*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)+x^8-x^7-13*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)+x^{8}-x^{7}-13 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(Or(n = 0,n = 1),1,1/40*(-4*5^(1/2)+20)*(3/2-1/2*5^(1/2))^(-n)+1/40*(4*5^(1/2)+20)*(3/2+1/2*5^(1/2))^(-n)-n^2+3/8*2^n*n+2*n-2^(-1+n)-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-4 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(4 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{40}-n^{2}+\frac{3 \,2^{n} n}{8}+\\2 n -2^{-1+n}-1 & \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) = 176
a(7) = 469
a(8) = 1201
a(n+4) = n^2-4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-4*n-2, n >= 9
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) = 176
a \! \left(7\right) = 469
a \! \left(8\right) = 1201
a \! \left(n +4\right) = n^{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-4 n -2, \quad n \geq 9
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21265/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[13,x]*F[3,x]
F[3,x] = F[1,x]+F[27,x]+F[4,x]
F[4,x] = F[13,x]*F[5,x]
F[5,x] = F[3,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[13,x]*F[8,x]
F[8,x] = F[25,x]+F[9,x]
F[9,x] = F[10,x]*F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]+F[22,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = F[15,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[13,x]*F[19,x]
F[22,x] = F[23,x]
F[23,x] = F[13,x]*F[24,x]
F[24,x] = F[1,x]+F[22,x]
F[25,x] = F[26,x]
F[26,x] = F[10,x]^2*F[13,x]*F[17,x]
F[27,x] = F[13,x]*F[28,x]
F[28,x] = F[29,x]+F[3,x]
F[29,x] = F[13,x]*F[30,x]
F[30,x] = F[31,x]+F[42,x]+F[50,x]
F[31,x] = F[24,x]+F[32,x]
F[32,x] = F[11,x]+F[33,x]
F[33,x] = F[34,x]+F[35,x]+F[37,x]
F[34,x] = 0
F[35,x] = F[13,x]*F[36,x]
F[36,x] = F[22,x]+F[33,x]
F[37,x] = F[13,x]*F[38,x]
F[38,x] = F[11,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[13,x]*F[41,x]
F[41,x] = F[22,x]+F[39,x]
F[42,x] = F[43,x]
F[43,x] = F[10,x]^2*F[13,x]*F[44,x]
F[44,x] = F[24,x]+F[45,x]
F[45,x] = F[11,x]+F[46,x]
F[46,x] = F[34,x]+F[47,x]+F[49,x]
F[47,x] = F[13,x]*F[48,x]
F[48,x] = F[22,x]+F[46,x]
F[49,x] = F[13,x]*F[45,x]
F[50,x] = F[51,x]
F[51,x] = F[10,x]*F[13,x]*F[44,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_{13}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{13}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{3}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{25}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{3}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{42}\! \left(x \right)+F_{50}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = 0
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{22}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{24}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{34}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{22}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{44}\! \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_13(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_27(x) + F_4(x))
Eq(F_4(x), F_13(x)*F_5(x))
Eq(F_5(x), F_3(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_10(x)*F_13(x)*F_8(x))
Eq(F_8(x), F_25(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_22(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), F_15(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_13(x)*F_19(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_13(x)*F_24(x))
Eq(F_24(x), F_1(x) + F_22(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_10(x)**2*F_13(x)*F_17(x))
Eq(F_27(x), F_13(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_3(x))
Eq(F_29(x), F_13(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_42(x) + F_50(x))
Eq(F_31(x), F_24(x) + F_32(x))
Eq(F_32(x), F_11(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_35(x) + F_37(x))
Eq(F_34(x), 0)
Eq(F_35(x), F_13(x)*F_36(x))
Eq(F_36(x), F_22(x) + F_33(x))
Eq(F_37(x), F_13(x)*F_38(x))
Eq(F_38(x), F_11(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_13(x)*F_41(x))
Eq(F_41(x), F_22(x) + F_39(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_10(x)**2*F_13(x)*F_44(x))
Eq(F_44(x), F_24(x) + F_45(x))
Eq(F_45(x), F_11(x) + F_46(x))
Eq(F_46(x), F_34(x) + F_47(x) + F_49(x))
Eq(F_47(x), F_13(x)*F_48(x))
Eq(F_48(x), F_22(x) + F_46(x))
Eq(F_49(x), F_13(x)*F_45(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_10(x)*F_13(x)*F_44(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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_and_col_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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, 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