0231_1203_1230_2301
Counting sequence:
1, 1, 2, 6, 20, 67, 220, 712, 2302, 7520, 24978, 84513, 291002, 1017450, 3603354, 12897758, 46573908, 169421079, 620158364, 2282282877, 8438582364, 31330224695, 116749916688, 436502881317, 1636888571312, 6155120878833, 23202749475260, 87667885966303, 331942810262728, 1259331071089013, 4786413494397852, 18223014563779532, 69489863917199446, 265381206151931596, 1014907978428608650, 3886466257793318553, 14901224686198724986, 57200225883494217682, 219813630523521842234, 845604293774048231907, 3256206921121979741666, 12550684541376559712902, 48418728900159638152858, 186952509369538362560791, 722443452346527980499446, 2793926351784049221234384, 10813083361227072790021958, 41878739812727224678419055, 162305773470316631379918790, 629445772925553180513435976, 2442615439510404305969416406, 9484477897164936911862677043, 36848777575020970090189887410, 143243336718460914114136542238, 557132339535764367764147762794, 2168035140449561443264832261233, 8440927249230315440079360313094, 32879272943780929438180507030344, 128131285467708322714995975315902, 499553709088765033919233634365315, 1948483209219144056449081536727666, 7603129681988481668063021174745286, 29679970175745309271480588957606442, 115905547115965034213530158472780446, 452803021778083677541739888655120596, 1769593977667500314514081395084561095, 6918186866171099748002091385601972764, 27055815583561422270056099240863735285, 105845940763673987017633343542286177612, 414217808997544941757515046299421926543, 1621512275999827894832645717093107835264, 6349576120726386459668225914906209625537, 24871301707559259835132020843445595384776, 97449140039272263357087369080284942226749, 381926497693855457034479156302065564430772, 1497271519746696695346130387281282463486887, 5871339211651552076754099263843371631303888, 23029609855390494092454590132110020437066473, 90353685617178048974780010560366407851721108, 354578372983246006665196423396108764655544845, 1391819743439286249472870253544666183589490428, 5464564097936891896032549894075659414206926095, 21459882312605380492520435700497153057493526272, 84293874530359190576940623171142213179764678431, 331176294206384862736340609477817036693713494316, 1301411895266812091392293996322842246745007023215, 5115174378943972854512170751754349773499292682024, 20109171129728899254586881568008666063518596006655, 79070399986501875263699746054650923915835141619812, 310969499008609420152622496944626673172183975685403, 1223218000097155282396019514533133218088347808875904, 4812495369146901993081782985362829110607721080455387, 18937184868832964563989762797073835522461860204696116, 74531085069869933652230524086462852276709889615793139, 293382870005155311672484591028902300237067070318051472, 1155063206828426870506072867110300001939368188192265917, 4548297791013942658087397780658465916076691898981561248, 17912767143524951164602446557895546855585653403683681433, 70557903181284888013458203991044100765261917173009897228, 277969049834192072548973937540882731727994819604893879855, 1095250968856191588610132850458499120024311812412775548648
Generating function in Maple syntax:
1/2*(9*(x^2-x+1/3)^2*(1-4*x)^(1/2)-13*x^4+20*x^3-15*x^2+6*x-1)/x/(2*x-1)/(x-1)^4
Generating function in latex syntax:
\frac{9 \left(x^{2}-x +\frac{1}{3}\right)^{2} \sqrt{1-4 x}-13 x^{4}+20 x^{3}-15 x^{2}+6 x -1}{2 x \left(2 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-13*x**4/2 + 10*x**3 - 15*x**2/2 + 3*x + 9*sqrt(1 - 4*x)*(x**2 - x + 1/3)**2/2 - 1/2)/(x*(x - 1)**4*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^8*F(x)^2+(2*x-1)*(13*x^4-20*x^3+15*x^2-6*x+1)*(x-1)^4*F(x)+81*x^8-302*x^7+545*x^6-599*x^5+432*x^4-208*x^3+65*x^2-12*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{8} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(13 x^{4}-20 x^{3}+15 x^{2}-6 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+81 x^{8}-302 x^{7}+545 x^{6}-599 x^{5}+432 x^{4}-208 x^{3}+65 x^{2}-12 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 220
a(n+5) = 12*(2*n+3)/(6+n)*a(n)-6*(11*n+18)/(6+n)*a(1+n)+(71*n+177)/(6+n)*a(n+2)-2*(19*n+69)/(6+n)*a(n+3)+2*(5*n+24)/(6+n)*a(n+4)+(3*n^2-17*n-12)/(6+n), 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) = 20
a \! \left(5\right) = 67
a \! \left(6\right) = 220
a \! \left(n +5\right) = \frac{12 \left(2 n +3\right) a \! \left(n \right)}{6+n}-\frac{6 \left(11 n +18\right) a \! \left(1+n \right)}{6+n}+\frac{\left(71 n +177\right) a \! \left(n +2\right)}{6+n}-\frac{2 \left(19 n +69\right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(5 n +24\right) a \! \left(n +4\right)}{6+n}+\frac{3 n^{2}-17 n -12}{6+n}, \quad n \geq 7
Specification 1
Strategy pack name: point_and_row_placements
Tree: http://permpal.com/tree/21452/
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[8,x]
F[3,x] = F[1,x]+F[34,x]+F[4,x]
F[4,x] = F[5,x]*F[8,x]
F[5,x] = F[3,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[18,x]+F[7,x]
F[10,x] = F[11,x]*F[14,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[8,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]^2*F[8,x]
F[18,x] = F[19,x]
F[19,x] = F[11,x]*F[20,x]*F[8,x]
F[20,x] = F[21,x]+F[29,x]
F[21,x] = F[14,x]*F[22,x]
F[22,x] = F[11,x]+F[23,x]
F[23,x] = F[12,x]+F[24,x]
F[24,x] = F[25,x]+F[26,x]+F[28,x]
F[25,x] = 0
F[26,x] = F[27,x]*F[8,x]
F[27,x] = F[12,x]+F[24,x]
F[28,x] = F[23,x]*F[8,x]
F[29,x] = F[12,x]+F[30,x]
F[30,x] = F[25,x]+F[31,x]+F[33,x]
F[31,x] = F[32,x]*F[8,x]
F[32,x] = F[12,x]+F[30,x]
F[33,x] = F[29,x]*F[8,x]
F[34,x] = F[35,x]*F[8,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[37,x]
F[37,x] = F[14,x]^2
F[38,x] = F[39,x]
F[39,x] = F[14,x]*F[40,x]*F[8,x]
F[40,x] = F[35,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[11,x]*F[43,x]*F[8,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[11,x]+F[29,x]
F[45,x] = F[46,x]
F[46,x] = F[11,x]^2*F[16,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_{8}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right) F_{8}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)+F_{7}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)
F_{21}\! \left(x \right) = F_{14}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{8}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{14} \left(x \right)^{2}
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right) F_{40}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{35}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right) F_{43}\! \left(x \right) F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{16}\! \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)*F_8(x))
Eq(F_3(x), F_1(x) + F_34(x) + F_4(x))
Eq(F_4(x), F_5(x)*F_8(x))
Eq(F_5(x), F_3(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_18(x) + F_7(x))
Eq(F_10(x), F_11(x)*F_14(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_8(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)**2*F_8(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_20(x)*F_8(x))
Eq(F_20(x), F_21(x) + F_29(x))
Eq(F_21(x), F_14(x)*F_22(x))
Eq(F_22(x), F_11(x) + F_23(x))
Eq(F_23(x), F_12(x) + F_24(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_28(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_27(x)*F_8(x))
Eq(F_27(x), F_12(x) + F_24(x))
Eq(F_28(x), F_23(x)*F_8(x))
Eq(F_29(x), F_12(x) + F_30(x))
Eq(F_30(x), F_25(x) + F_31(x) + F_33(x))
Eq(F_31(x), F_32(x)*F_8(x))
Eq(F_32(x), F_12(x) + F_30(x))
Eq(F_33(x), F_29(x)*F_8(x))
Eq(F_34(x), F_35(x)*F_8(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_14(x)**2)
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_14(x)*F_40(x)*F_8(x))
Eq(F_40(x), F_35(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_11(x)*F_43(x)*F_8(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_11(x) + F_29(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_11(x)**2*F_16(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"}, {"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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_and_row_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, 0, 3], "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], "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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, 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