0132_0213_0231_1032_3012
Counting sequence:
1, 1, 2, 6, 19, 56, 162, 464, 1319, 3726, 10469, 29278, 81546, 226306, 626029, 1726806, 4750807, 13039844, 35714994, 97629716, 266401619, 725732958, 1974043241, 5361976906, 14545381914, 39409072006, 106653049177, 288328817574, 778698414619, 2101082717456, 5664133600194, 15256753379288, 41062848563711, 110436523093806, 296804191037549, 797143730196406, 2139572569764522, 5739243009558154, 15386217980080069, 41225926463731446, 110403046489092319, 295511152704430268, 790602745648874706, 2114186146615337756, 5651150547291894059, 15098960992171467006, 40325624066861119217, 107657890624416604834, 287308094416764225594, 766466553040997948686, 2044031999341219720369, 5449250588472509635398, 14522642761910864354851, 38691825541273900513256, 103053354398117733268194, 274396651417686170457056, 730421320612554495516887, 1943793058928211597169614, 5171450380939169421806069, 13755109909682329764614926, 36577042300719830035323594, 97240954024520157734844178, 258457467916357625186389021, 686801457123060666482374806, 1824645277504831238217711079, 4846559490148945691383633556, 12870610163162536901614839282, 34172576795242745567292958244, 90713460640057410396078607619, 240759520581500536854554507166, 638873907056665643272605269369, 1694996902988589671044710724858, 4496212103157161761000854821466, 11924790607826977507294388064406, 31621518023107958055752584430089, 83838687168505376649237656076006, 222247956300649424564912981290027, 589066496481158175075084789420224, 1561082064567729682076139294356994, 4136409976749029337371143314183656, 10958708172835449147274923402406319, 29029165183698592089948714935139438, 76886578996928058261818852775112061, 203614496021147502127755649464392806, 539150890090032475278943882468554474, 1427436193105442085749315552418538906, 3778757764772252240932225651368378997, 10002039308992697851896790476953280054, 26471366710004032500343210128710190319, 70050878256632589991038603881566796396, 185353713818935117312904829083939986194, 490388783041683711126166802101713018668, 1297275749071607483760936105847628851931, 3431449285628102194064172182589003027006, 9075641458412097992458831914736438916609, 24001172319951425011486547312923660295506, 63466397841872477832496229805127523000442, 167807890996613677629314177694434505224734, 443648362180381261694886990273009801199969, 1172800586850820558230356818517120727434886, 3100052485258539058681772854461096059757619
Generating function in Maple syntax:
(x-1)*(x^5-2*x^4-2*x^3-3*x^2+4*x-1)/(x^2-3*x+1)^2
Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{5}-2 x^{4}-2 x^{3}-3 x^{2}+4 x -1\right)}{\left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(x - 1)*(x**5 - 2*x**4 - 2*x**3 - 3*x**2 + 4*x - 1)/(x**2 - 3*x + 1)**2
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)^2*F(x)+(x-1)*(x^5-2*x^4-2*x^3-3*x^2+4*x-1) = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+\left(x -1\right) \left(x^{5}-2 x^{4}-2 x^{3}-3 x^{2}+4 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,1/25*((5*n+36)*5^(1/2)-10*n-75)*(3/2-1/2*5^(1/2))^(-n)-1/5*((n+36/5)*5^(1/2)+2*n+15)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(5 n +36\right) \sqrt{5}-10 n -75\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{25}-\\\frac{\left(\left(n +\frac{36}{5}\right) \sqrt{5}+2 n +15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{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) = 19
a(5) = 56
a(6) = 162
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3), 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) = 56
a \! \left(6\right) = 162
a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18152/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[18,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[1,x]+F[4,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]+F[39,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[22,x]*F[4,x]
F[25,x] = F[19,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[38,x]
F[27,x] = 0
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]+F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[23,x]*F[4,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[27,x]+F[28,x]+F[35,x]
F[35,x] = F[19,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[26,x]*F[4,x]
F[38,x] = F[25,x]*F[4,x]
F[39,x] = F[27,x]+F[40,x]+F[64,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[48,x]
F[42,x] = F[11,x]+F[43,x]
F[43,x] = F[27,x]+F[44,x]+F[46,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[11,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[23,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[27,x]+F[40,x]+F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[19,x]+F[49,x]
F[52,x] = F[27,x]+F[53,x]+F[60,x]+F[62,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[56,x]
F[56,x] = F[11,x]*F[4,x]
F[57,x] = F[58,x]
F[58,x] = 2*F[27,x]+F[53,x]+F[59,x]
F[59,x] = F[4,x]*F[49,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[49,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[26,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[51,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[27,x]+F[28,x]+F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[19,x]+F[34,x]
F[71,x] = F[67,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{18}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \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_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{38}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{27}\! \left(x \right)+F_{40}\! \left(x \right)+F_{64}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{27}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{23}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{27}\! \left(x \right)+F_{40}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{27}\! \left(x \right)+F_{53}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{53}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{49}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{26}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{51}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)
F_{71}\! \left(x \right) = F_{67}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_18(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(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_4(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_1(x) + F_4(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x) + F_39(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_22(x)*F_4(x))
Eq(F_25(x), F_19(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_38(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_31(x) + F_4(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_23(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_27(x) + F_28(x) + F_35(x))
Eq(F_35(x), F_19(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_26(x)*F_4(x))
Eq(F_38(x), F_25(x)*F_4(x))
Eq(F_39(x), F_27(x) + F_40(x) + F_64(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_48(x))
Eq(F_42(x), F_11(x) + F_43(x))
Eq(F_43(x), F_27(x) + F_44(x) + F_46(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_11(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_23(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_27(x) + F_40(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_19(x) + F_49(x))
Eq(F_52(x), F_27(x) + F_53(x) + F_60(x) + F_62(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_11(x)*F_4(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), 2*F_27(x) + F_53(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_49(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_49(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_26(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_51(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_27(x) + F_28(x) + F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_19(x) + F_34(x))
Eq(F_71(x), F_67(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:
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