0123_0231_0312_1230
Counting sequence:
1, 1, 2, 6, 20, 69, 242, 858, 3068, 11050, 40052, 145996, 534888, 1968685, 7276050, 26993490, 100490220, 375287550, 1405622460, 5278838100, 19873977240, 74994427170, 283595947284, 1074568266756, 4079184055640, 15511924233204, 59083160374952, 225384613313944, 861009442488272, 3293658897463165, 12615466728509778, 48378415060957954, 185735780008951116, 713853847957350646, 2746432574830171500, 10576752760809133444, 40769839031961545912, 157293137880435964246, 607361006504615712860, 2347111909451832900300, 9077263714458819858440, 35131433924019168359820, 136063922420273062282200, 527331292525602707544360, 2045057773596167869383600, 7935930327367345619259090, 30814118059537871492459220, 119715747239703863688358932, 465364270006369563317307768, 1809948020208142322954152620, 7043070894740003023714250712, 27420346354749195990907440456, 106804899987914238245963795952, 416207230320134722204257221796, 1622637333521411899370176998120, 6328783611830633398016506559304, 24694476573402390406716069018672, 96394965398187177213631051769448, 376424825242301315898192280818768, 1470504080710683422662563504529200, 5746629767090684394510226273563552, 22465411420928895087037874150152221, 87854718612911247630113337113119698, 343685295882740950108195263175198050, 1344923971459900697419992299727871500, 5264672976318945198872849367635830950, 20614750249375466812727093030869889100, 80744706602200902505475441256481990500, 316355794375259511167549566024799479800, 1239823501005840379686306659054382036750, 4860299124806106707213006501035296338700, 19058207504824200627168004135310967647100, 74750482409631409224419335262430622592040, 293262098646275038643448506462122670220060, 1150813825886201948563488486438252473969400, 4517089623078184177212859333539935872316040, 17734308425746135804158251571876606310433520, 69641677636374603916368683727251631080049990, 273540227692406507953258100305293704567387420, 1074652062900034992553845173605426425886767900, 4222863127452839992608706835909792729068071080, 16597261270266369298800455153030557331398887940, 65246026570741446536030817222156600394315393160, 256541663295814142690019013025476781612359107160, 1008895066025009862737119049153675889219022818000, 3968409352128423202902508280253162144544998755740, 15612321541327306120278840955778266722926178938520, 61432375299610846230243470677308465809291592562360, 241771009927403253315572037365760001866997742888400, 951670463598588743796070333732548640670376808327000, 3746647291357289793174509674871112650367648767543600, 14752694179708876864620670016769674003010131116190800, 58099270666682899791578830155271500366457636369820000, 228843922046481850182793018284351300379986363911737650, 901521394568549574829110166053395164385706974391996500, 3552051730415470726171819162959804510223851194324594100, 13997413462706873576375671386931711926006938610672367800, 55167115940342220281136249350145559666810575906113623740, 217458056191619529168642710923516641932753717631608163960, 857298855701027310606856392754867143328823346786237959400, 3380253963842974380998684638851634708876281734550999710640
Generating function in Maple syntax:
1/2*(-4*(x-1/2)^2*(-4*x+1)^(1/2)-2*x^3+10*x^2-6*x+1)/x^3
Generating function in latex syntax:
\frac{-4 \left(x -\frac{1}{2}\right)^{2} \sqrt{-4 x +1}-2 x^{3}+10 x^{2}-6 x +1}{2 x^{3}}
Generating function in sympy syntax:
(-x**3 + 5*x**2 - 3*x - 2*sqrt(1 - 4*x)*(x - 1/2)**2 + 1/2)/x**3
Implicit equation for the generating function in Maple syntax:
x^3*F(x)^2+(2*x^3-10*x^2+6*x-1)*F(x)+x^3+6*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{3} F \! \left(x \right)^{2}+\left(2 x^{3}-10 x^{2}+6 x -1\right) F \! \left(x \right)+x^{3}+6 x^{2}-5 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+2) = -4*(1+2*n)/(5+n)*a(n)+6*(3+n)/(5+n)*a(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) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{5+n}+\frac{6 \left(3+n \right) a \! \left(n +1\right)}{5+n}, \quad n \geq 3
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21150/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[12,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[56,x]
F[4,x] = F[12,x]*F[5,x]
F[5,x] = F[1,x]+F[54,x]+F[55,x]+F[6,x]
F[6,x] = F[7,x,1]
F[7,x,k[0]] = F[8,x,k[0]]*F[9,x,k[0]]
F[8,x,k[0]] = k[0]*x
F[9,x,k[0]] = F[1,x]+F[10,x,k[0]]+F[13,x,k[0]]+F[7,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]*F[12,x]
F[11,x,k[0]] = (F[9,x,k[0]]*k[0]-F[9,x,1])/(-1+k[0])
F[12,x] = F[8,x,1]
F[13,x,k[0]] = F[12,x]*F[14,x,k[0]]
F[14,x,k[0]] = F[1,x]+F[13,x,k[0]]+F[15,x,k[0]]+F[16,x]+F[53,x]
F[15,x,k[0]] = F[14,x,k[0]]*F[8,x,k[0]]
F[16,x] = F[17,x,1]
F[17,x,k[0]] = F[12,x]*F[18,x,k[0]]
F[18,x,k[0]] = F[1,x]+F[19,x,k[0]]+F[48,x,k[0]]+F[50,x,k[0]]+F[51,x]
F[19,x,k[0]] = F[12,x]*F[20,x,k[0]]
F[20,x,k[0]] = F[21,x,k[0]]+F[31,x,k[0]]
F[21,x,k[0]] = F[22,x,k[0]]+F[25,x,k[0]]
F[22,x,k[0]] = F[1,x]+F[23,x,k[0]]
F[23,x,k[0]] = F[24,x,k[0]]
F[24,x,k[0]] = F[22,x,k[0]]*F[8,x,k[0]]
F[25,x,k[0]] = F[26,x]+F[29,x,k[0]]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[1,x]+F[26,x]
F[29,x,k[0]] = F[30,x,k[0]]
F[30,x,k[0]] = F[25,x,k[0]]*F[8,x,k[0]]
F[31,x,k[0]] = F[32,x,k[0]]+F[38,x,k[0]]
F[32,x,k[0]] = F[26,x]+F[33,x,k[0]]
F[33,x,k[0]] = F[34,x]+F[35,x,k[0]]+F[37,x,k[0]]
F[34,x] = 0
F[35,x,k[0]] = F[12,x]*F[36,x,k[0]]
F[36,x,k[0]] = F[23,x,k[0]]+F[33,x,k[0]]
F[37,x,k[0]] = F[32,x,k[0]]*F[8,x,k[0]]
F[38,x,k[0]] = F[39,x]+F[44,x,k[0]]
F[39,x] = F[34,x]+F[40,x]+F[42,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[26,x]+F[39,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[26,x]+F[39,x]
F[44,x,k[0]] = 2*F[34,x]+F[45,x,k[0]]+F[47,x,k[0]]
F[45,x,k[0]] = F[12,x]*F[46,x,k[0]]
F[46,x,k[0]] = F[29,x,k[0]]+F[44,x,k[0]]
F[47,x,k[0]] = F[38,x,k[0]]*F[8,x,k[0]]
F[48,x,k[0]] = F[12,x]*F[49,x,k[0]]
F[49,x,k[0]] = 1/(-1+k[0])*(F[18,x,k[0]]*k[0]-F[18,x,1])
F[50,x,k[0]] = F[18,x,k[0]]*F[8,x,k[0]]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[28,x]+F[43,x]
F[53,x] = F[12,x]*F[52,x]
F[54,x] = F[10,x,1]
F[55,x] = F[13,x,1]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[1,x]+F[16,x]+F[53,x]+F[56,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_{12}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{56}\! \left(x \right)
F_{4}\! \left(x \right) = F_{12}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x , 1\right)
F_{7}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = y x
F_{9}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{12}\! \left(x \right)
F_{11}\! \left(x , y\right) = \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}
F_{12}\! \left(x \right) = F_{8}\! \left(x , 1\right)
F_{13}\! \left(x , y\right) = F_{12}\! \left(x \right) F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{16}\! \left(x \right)+F_{53}\! \left(x \right)
F_{15}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{16}\! \left(x \right) = F_{17}\! \left(x , 1\right)
F_{17}\! \left(x , y\right) = F_{12}\! \left(x \right) F_{18}\! \left(x , y\right)
F_{18}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{48}\! \left(x , y\right)+F_{50}\! \left(x , y\right)+F_{51}\! \left(x \right)
F_{19}\! \left(x , y\right) = F_{12}\! \left(x \right) F_{20}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right)+F_{31}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)+F_{25}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{22}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{26}\! \left(x \right)+F_{29}\! \left(x , y\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_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x , y\right) = F_{30}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{25}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{31}\! \left(x , y\right) = F_{32}\! \left(x , y\right)+F_{38}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{26}\! \left(x \right)+F_{33}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{34}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{37}\! \left(x , y\right)
F_{34}\! \left(x \right) = 0
F_{35}\! \left(x , y\right) = F_{12}\! \left(x \right) F_{36}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = F_{23}\! \left(x , y\right)+F_{33}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = F_{32}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{38}\! \left(x , y\right) = F_{39}\! \left(x \right)+F_{44}\! \left(x , y\right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{26}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{26}\! \left(x \right)+F_{39}\! \left(x \right)
F_{44}\! \left(x , y\right) = 2 F_{34}\! \left(x \right)+F_{45}\! \left(x , y\right)+F_{47}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{12}\! \left(x \right) F_{46}\! \left(x , y\right)
F_{46}\! \left(x , y\right) = F_{29}\! \left(x , y\right)+F_{44}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{38}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{12}\! \left(x \right) F_{49}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = \frac{y F_{18}\! \left(x , y\right)-F_{18}\! \left(x , 1\right)}{-1+y}
F_{50}\! \left(x , y\right) = F_{18}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{28}\! \left(x \right)+F_{43}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x , 1\right)
F_{55}\! \left(x \right) = F_{13}\! \left(x , 1\right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{53}\! \left(x \right)+F_{56}\! \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_12(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_56(x))
Eq(F_4(x), F_12(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_54(x) + F_55(x) + F_6(x))
Eq(F_6(x), F_7(x, 1))
Eq(F_7(x, k_0), F_8(x, k_0)*F_9(x, k_0))
Eq(F_8(x, k_0), k_0*x)
Eq(F_9(x, k_0), F_1(x) + F_10(x, k_0) + F_13(x, k_0) + F_7(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0)*F_12(x))
Eq(F_11(x, k_0), (-k_0*F_9(x, k_0) + F_9(x, 1))/(1 - k_0))
Eq(F_12(x), F_8(x, 1))
Eq(F_13(x, k_0), F_12(x)*F_14(x, k_0))
Eq(F_14(x, k_0), F_1(x) + F_13(x, k_0) + F_15(x, k_0) + F_16(x) + F_53(x))
Eq(F_15(x, k_0), F_14(x, k_0)*F_8(x, k_0))
Eq(F_16(x), F_17(x, 1))
Eq(F_17(x, k_0), F_12(x)*F_18(x, k_0))
Eq(F_18(x, k_0), F_1(x) + F_19(x, k_0) + F_48(x, k_0) + F_50(x, k_0) + F_51(x))
Eq(F_19(x, k_0), F_12(x)*F_20(x, k_0))
Eq(F_20(x, k_0), F_21(x, k_0) + F_31(x, k_0))
Eq(F_21(x, k_0), F_22(x, k_0) + F_25(x, k_0))
Eq(F_22(x, k_0), F_1(x) + F_23(x, k_0))
Eq(F_23(x, k_0), F_24(x, k_0))
Eq(F_24(x, k_0), F_22(x, k_0)*F_8(x, k_0))
Eq(F_25(x, k_0), F_26(x) + F_29(x, k_0))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_1(x) + F_26(x))
Eq(F_29(x, k_0), F_30(x, k_0))
Eq(F_30(x, k_0), F_25(x, k_0)*F_8(x, k_0))
Eq(F_31(x, k_0), F_32(x, k_0) + F_38(x, k_0))
Eq(F_32(x, k_0), F_26(x) + F_33(x, k_0))
Eq(F_33(x, k_0), F_34(x) + F_35(x, k_0) + F_37(x, k_0))
Eq(F_34(x), 0)
Eq(F_35(x, k_0), F_12(x)*F_36(x, k_0))
Eq(F_36(x, k_0), F_23(x, k_0) + F_33(x, k_0))
Eq(F_37(x, k_0), F_32(x, k_0)*F_8(x, k_0))
Eq(F_38(x, k_0), F_39(x) + F_44(x, k_0))
Eq(F_39(x), F_34(x) + F_40(x) + F_42(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_26(x) + F_39(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_26(x) + F_39(x))
Eq(F_44(x, k_0), 2*F_34(x) + F_45(x, k_0) + F_47(x, k_0))
Eq(F_45(x, k_0), F_12(x)*F_46(x, k_0))
Eq(F_46(x, k_0), F_29(x, k_0) + F_44(x, k_0))
Eq(F_47(x, k_0), F_38(x, k_0)*F_8(x, k_0))
Eq(F_48(x, k_0), F_12(x)*F_49(x, k_0))
Eq(F_49(x, k_0), (k_0*F_18(x, k_0) - F_18(x, 1))/(k_0 - 1))
Eq(F_50(x, k_0), F_18(x, k_0)*F_8(x, k_0))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_28(x) + F_43(x))
Eq(F_53(x), F_12(x)*F_52(x))
Eq(F_54(x), F_10(x, 1))
Eq(F_55(x), F_13(x, 1))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_1(x) + F_16(x) + F_53(x) + F_56(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_placements_tracked_fusion", "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:
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