0231_1230_2301
Counting sequence:
1, 1, 2, 6, 21, 77, 285, 1055, 3905, 14476, 53812, 200709, 751206, 2820944, 10625962, 40138957, 152012381, 577043255, 2195164166, 8367157537, 31950184190, 122206206340, 468149252841, 1795969822629, 6899137076491, 26535865305598, 102183297396743, 393915846158838, 1520111466483996, 5871782384226937, 22701958348378731, 87848398707670920, 340221378496396880, 1318645986609055269, 5114659885521824834, 19852333125251851404, 77107757963431539496, 299683500256327589874, 1165448997947273567911, 4535004224377608194413, 17656522723544219175660, 68780527424516479256085, 268070546272857743604391, 1045314548273131230146677, 4078033198298423037411031, 15916678920661271991647850, 62150499262123975550731672, 242784083872629376038887247, 948795416951849366642606486, 3709329677074180136423131316, 14507174443891794784455441132, 56758269063283131758747734440, 222141152256209862989122456551, 869715874356956182460073330089, 3406194603848220941621858885082, 13344448168104819496117074794201, 52295765679897122308944506284087, 205004325902012545886128186784341, 803870625605303233444378385078447, 3153059392746646510306564297619924, 12370788904842865347963127869305942, 48548794241642713230145953898303981, 190577670128256641802409638726256676, 748298610660523316457526173833775373, 2938895924785898129879686606345345398, 11545079409665424303242973271675646976, 45363893201680959178215458134054808407, 178287843165840943217242499934972398419, 700855694337096978513224279815197523783, 2755678145529195824961695502617697319104, 10837248480200366470267412983864696990825, 42628294472110440259051510431275688667266, 167711595091259052066898489787965549468910, 659951969217621030544615634699943880227245, 2597428816758710044271294669104909652305215, 10224806966815194600682396036289495050968543, 40257318291061918702121417797188013555779366, 158529837418798921890961887266994729542236337, 624384177420772895827171267151608784938026036, 2459607331632883164641061528546343229645728664, 9690608643247521315305539440245622794882440925, 38186173632874139966919741832796998602714678553, 150497581871010503894365970804631105360006745609, 593225376696562154377359661786138429030086371820, 2338704313114691342778629162782274597393278085847, 9221359050299248081334325178140833567950757459768, 36364467733989657944839918150174891934814877579918, 143423703962883415779336092736126621625618456852279, 565750281280999447513074946592827056298505617556657, 2231965907377363005594079012709267369803437134856241, 8806596398926624912269779141177494602691062638627048, 34752435775838498832799738214241416279950418930902805, 137156981742608574385376137151117094371985062570245144, 541383519535952804589184902478147445446768417239684264, 2137201857823936664050418055814681351284754114033534853, 8437979080725523796482868216661034705150007989249659979, 33318293775007393698315519693788310721020923931184573777, 131576253275963286275436494052177432173211851558475050432, 519663009330669698142366318822842445555316499704117321511, 2052649120467147960777998695310071332876926672826643068968, 8108777419924874623760458956938866360972820732953343698708
Generating function in Maple syntax:
-1/2*(6*(1-4*x)^(1/2)*x^3+18*x^4-9*(1-4*x)^(1/2)*x^2-38*x^3+5*(1-4*x)^(1/2)*x+29*x^2-(1-4*x)^(1/2)-9*x+1)*(2*x-1)/x/(9*x^3-14*x^2+7*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{\left(6 \sqrt{1-4 x}\, x^{3}+18 x^{4}-9 \sqrt{1-4 x}\, x^{2}-38 x^{3}+5 \sqrt{1-4 x}\, x +29 x^{2}-\sqrt{1-4 x}-9 x +1\right) \left(2 x -1\right)}{2 x \left(9 x^{3}-14 x^{2}+7 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(2*x - 1)*(-9*x**4 - 3*x**3*sqrt(1 - 4*x) + 19*x**3 + 9*x**2*sqrt(1 - 4*x)/2 - 29*x**2/2 - 5*x*sqrt(1 - 4*x)/2 + 9*x/2 + sqrt(1 - 4*x)/2 - 1/2)/(x*(x - 1)**2*(9*x**3 - 14*x**2 + 7*x - 1))
Implicit equation for the generating function in Maple syntax:
x*(9*x^3-14*x^2+7*x-1)*(x-1)^4*F(x)^2+(2*x-1)*(18*x^4-38*x^3+29*x^2-9*x+1)*(x-1)^2*F(x)+(9*x^4-20*x^3+18*x^2-7*x+1)*(2*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
x \left(9 x^{3}-14 x^{2}+7 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(18 x^{4}-38 x^{3}+29 x^{2}-9 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(9 x^{4}-20 x^{3}+18 x^{2}-7 x +1\right) \left(2 x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(6) = 285
a(7) = 1055
a(8) = 3905
a(n+8) = -108*(2*n+3)/(n+9)*a(n)+6*(155*n+336)/(n+9)*a(1+n)-3*(577*n+1720)/(n+9)*a(n+2)+(7105+1822*n)/(n+9)*a(n+3)-5*(237*n+1159)/(n+9)*a(n+4)+2*(243*n+1439)/(n+9)*a(n+5)-2*(61*n+425)/(n+9)*a(n+6)+17*(n+8)/(n+9)*a(n+7)+2/(n+9), 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) = 21
a \! \left(5\right) = 77
a \! \left(6\right) = 285
a \! \left(7\right) = 1055
a \! \left(8\right) = 3905
a \! \left(n +8\right) = -\frac{108 \left(2 n +3\right) a \! \left(n \right)}{n +9}+\frac{6 \left(155 n +336\right) a \! \left(1+n \right)}{n +9}-\frac{3 \left(577 n +1720\right) a \! \left(n +2\right)}{n +9}+\frac{\left(7105+1822 n \right) a \! \left(n +3\right)}{n +9}-\frac{5 \left(237 n +1159\right) a \! \left(n +4\right)}{n +9}+\frac{2 \left(243 n +1439\right) a \! \left(n +5\right)}{n +9}-\frac{2 \left(61 n +425\right) a \! \left(n +6\right)}{n +9}+\frac{17 \left(n +8\right) a \! \left(n +7\right)}{n +9}+\frac{2}{n +9}, \quad n \geq 9
Specification 1
Strategy pack name: insertion_point_col_placements
Tree: http://permpal.com/tree/367/
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[6,x]+F[9,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[6,x]^2*F[4,x]
F[9,x] = F[10,x]+F[2,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[4,x]*F[6,x]
F[12,x] = F[13,x]+F[23,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,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[20,x]*F[4,x]
F[20,x] = F[17,x]+F[7,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[4,x]*F[6,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[2,x]*F[6,x]
F[25,x] = F[26,x]+F[36,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[34,x]
F[29,x] = F[14,x]+F[30,x]
F[30,x] = F[18,x]+F[31,x]+F[33,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[14,x]+F[30,x]
F[33,x] = F[29,x]*F[4,x]
F[34,x] = F[2,x]*F[35,x]
F[35,x] = F[16,x]+F[29,x]
F[36,x] = F[18,x]+F[37,x]+F[51,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[49,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[2,x]*F[7,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[40,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[14,x]*F[47,x]*F[7,x]
F[47,x] = F[16,x]+F[48,x]
F[48,x] = F[2,x]+F[26,x]
F[49,x] = F[14,x]*F[50,x]
F[50,x] = F[20,x]+F[40,x]
F[51,x] = F[52,x]
F[52,x] = F[25,x]*F[4,x]*F[6,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_{6}\! \left(x \right)+F_{9}\! \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_{6} \left(x \right)^{2} F_{4}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{2}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{23}\! \left(x \right)
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_{16}\! \left(x \right) F_{4}\! \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_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right) F_{6}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right)+F_{29}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)+F_{51}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right) F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{47}\! \left(x \right) F_{7}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{26}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)+F_{40}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \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_6(x) + F_9(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_6(x)**2)
Eq(F_9(x), F_10(x) + F_2(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_4(x)*F_6(x))
Eq(F_12(x), F_13(x) + F_23(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(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_20(x)*F_4(x))
Eq(F_20(x), F_17(x) + F_7(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_4(x)*F_6(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_2(x)*F_6(x))
Eq(F_25(x), F_26(x) + F_36(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_34(x))
Eq(F_29(x), F_14(x) + F_30(x))
Eq(F_30(x), F_18(x) + F_31(x) + F_33(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_14(x) + F_30(x))
Eq(F_33(x), F_29(x)*F_4(x))
Eq(F_34(x), F_2(x)*F_35(x))
Eq(F_35(x), F_16(x) + F_29(x))
Eq(F_36(x), F_18(x) + F_37(x) + F_51(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_49(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_2(x)*F_7(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_40(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_14(x)*F_47(x)*F_7(x))
Eq(F_47(x), F_16(x) + F_48(x))
Eq(F_48(x), F_2(x) + F_26(x))
Eq(F_49(x), F_14(x)*F_50(x))
Eq(F_50(x), F_20(x) + F_40(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_25(x)*F_4(x)*F_6(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": false, "strategy_class": "RowAndColumnPlacementFactory"}], [{"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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_point_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, 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, 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, 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, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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