0132_0231_0312_1203_3012
Counting sequence:
1, 1, 2, 6, 19, 58, 175, 538, 1709, 5612, 18944, 65315, 228823, 811608, 2907160, 10498204, 38171707, 139617396, 513320463, 1895965065, 7031558519, 26174201312, 97756061141, 366211168843, 1375699949179, 5181088897066, 19558540580399, 73993381774859, 280492870718351, 1065275867979728, 4052833280440975, 15444089041414902, 58942428118704961, 225276205329900832, 862158763640051284, 3303771996204124267, 12675144128947039627, 48684131142380503996, 187192112378126503880, 720492104044812415545, 2775813239841689957317, 10704096522343396986766, 41313258500733454737772, 159584531076390848546005, 616932299662139311654575, 2386789829462500554428652, 9240732390760805931770226, 35801454409788323836744397, 138798372352741793408295005, 538450329159381639149698562, 2090130241700136612568280998, 8118148616443495014687191797, 31549081332813587428805546251, 122674189237923508376774817624, 477251871704559054024626436144, 1857640786170402049778560309215, 7234151912228878486508653241081, 28184987791893607514994173664842, 109861457264471842546299945842402, 428413784114081916391955792113477, 1671343536302908403613494448987031, 6522981924748381212700049308711876, 25468245331557246078866093583888912, 99476109578274071983008461437954356, 388687211674287482278785771477400635, 1519282100358718985177275380585074876, 5940575821648953792729470334505922543, 23236251885114483654955101091914190025, 90917330669243950257450361914034253911, 355849363018109914181894736323382811240, 1393223048667655778974806813058356551101, 5456400546405172939267381057444805073183, 21375649773132936396939537050522858639475, 83763805177119002063964036158518116582666, 328332676730163600394824093741801115274435, 1287328050536014662140499851376794741714771, 5048689717501305975229836805786999459299743, 19805211254903380542890660594327697732126280, 77712122441512291880817074968515429998428051, 305002785009432933649338192062116826910706255, 1197352256356175274567115830394619111048795111, 4701545231022026584011365579865063022883324438, 18465343686980524427928114234057791972734381925, 72538722319781472719649711549615461466581803091, 285020341854932050402551514525654906583681934895, 1120141943582451958504061816999866204575931595556, 4403108498513437189557083872955862168963033598153, 17311423823646549451565063194884533623136867685325, 68075569766693379651949210396555592220524356233751, 267752146626559918154941445864103741794016044433910, 1053309421370641481724646263267598299997189271016181, 4144369974250300407654724604069664861082415911703083, 16309433664148658915070781900826746638597678208843535, 64194139352728661909290556437560875922648316983199056, 252712310113368273627143109113368765168865537730901765, 995016682902716476455208518954786044810538605971214431, 3918372808498596480099749220509986352235330160997377123, 15433026404030268036388773785907226508602342799339781310, 60794595009731565626264801422223626512991219675742092311, 239522315003197259937001350626313131172426800418959930535, 943827850228861594463403289036040188269299747383549731351
Generating function in Maple syntax:
-1/2*(-1+(1-4*x)^(1/2))/x*(x^4-3*x^3+6*x^2-4*x+1)/(x-1)^4
Generating function in latex syntax:
-\frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{4}-3 x^{3}+6 x^{2}-4 x +1\right)}{2 x \left(x -1\right)^{4}}
Generating function in sympy syntax:
(1/2 - sqrt(1 - 4*x)/2)*(x**4 - 3*x**3 + 6*x**2 - 4*x + 1)/(x*(x - 1)**4)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^8*F(x)^2-(x^4-3*x^3+6*x^2-4*x+1)*(x-1)^4*F(x)+(x^4-3*x^3+6*x^2-4*x+1)^2 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{8} F \! \left(x \right)^{2}-\left(x^{4}-3 x^{3}+6 x^{2}-4 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+\left(x^{4}-3 x^{3}+6 x^{2}-4 x +1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 58
a(6) = 175
a(7) = 538
a(8) = 1709
a(n+6) = -2*(2*n+1)/(n+7)*a(n)+(22+17*n)/(n+7)*a(n+1)-(40*n+113)/(n+7)*a(n+2)+(179+49*n)/(n+7)*a(n+3)-10*(3*n+14)/(n+7)*a(n+4)+(52+9*n)/(n+7)*a(n+5)+1/2*(n+1)*(n+2)/(n+7), 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) = 19
a \! \left(5\right) = 58
a \! \left(6\right) = 175
a \! \left(7\right) = 538
a \! \left(8\right) = 1709
a \! \left(n +6\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{\left(22+17 n \right) a \! \left(n +1\right)}{n +7}-\frac{\left(40 n +113\right) a \! \left(n +2\right)}{n +7}+\frac{\left(179+49 n \right) a \! \left(n +3\right)}{n +7}-\frac{10 \left(3 n +14\right) a \! \left(n +4\right)}{n +7}+\frac{\left(52+9 n \right) a \! \left(n +5\right)}{n +7}+\frac{\left(n +1\right) \left(n +2\right)}{2 n +14}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/21656/
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[17,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[17,x]*F[7,x]
F[7,x] = F[18,x]+F[8,x]
F[8,x] = F[9,x,1]
F[9,x,k[0]] = F[10,x,k[0]]+F[14,x,k[0]]
F[10,x,k[0]] = F[0,x]+F[11,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]*F[9,x,k[0]]
F[13,x,k[0]] = k[0]*x
F[14,x,k[0]] = F[15,x,k[0]]
F[15,x,k[0]] = F[16,x,k[0]]*F[17,x]
F[16,x,k[0]] = (F[9,x,k[0]]*k[0]-F[9,x,1])/(-1+k[0])
F[17,x] = x
F[18,x] = F[19,x]+F[32,x]
F[19,x] = F[20,x]
F[20,x] = F[17,x]*F[21,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[17,x]*F[22,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]
F[27,x] = F[17,x]*F[28,x]
F[28,x] = F[1,x]+F[26,x]
F[29,x] = F[30,x]
F[30,x] = F[17,x]*F[31,x]
F[31,x] = F[23,x]+F[29,x]
F[32,x] = F[33,x]+F[34,x]+F[35,x]
F[33,x] = 0
F[34,x] = F[17,x]*F[18,x]
F[35,x] = F[17,x]*F[36,x]
F[36,x] = F[25,x]+F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[39,x]
F[39,x] = F[17,x]*F[40,x]
F[40,x] = F[26,x]+F[38,x]
F[41,x] = F[42,x]
F[42,x] = F[17,x]*F[43,x]
F[43,x] = F[29,x]+F[41,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_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{18}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x , 1\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = y x
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right) F_{17}\! \left(x \right)
F_{16}\! \left(x , y\right) = \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}
F_{17}\! \left(x \right) = x
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{32}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right) F_{21}\! \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_{17}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \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 \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{17}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = 0
F_{34}\! \left(x \right) = F_{17}\! \left(x \right) F_{18}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{17}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{26}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{29}\! \left(x \right)+F_{41}\! \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_17(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_17(x)*F_7(x))
Eq(F_7(x), F_18(x) + F_8(x))
Eq(F_8(x), F_9(x, 1))
Eq(F_9(x, k_0), F_10(x, k_0) + F_14(x, k_0))
Eq(F_10(x, k_0), F_0(x) + F_11(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0)*F_9(x, k_0))
Eq(F_13(x, k_0), k_0*x)
Eq(F_14(x, k_0), F_15(x, k_0))
Eq(F_15(x, k_0), F_16(x, k_0)*F_17(x))
Eq(F_16(x, k_0), (-k_0*F_9(x, k_0) + F_9(x, 1))/(1 - k_0))
Eq(F_17(x), x)
Eq(F_18(x), F_19(x) + F_32(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_17(x)*F_21(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_17(x)*F_22(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_17(x)*F_28(x))
Eq(F_28(x), F_1(x) + F_26(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_17(x)*F_31(x))
Eq(F_31(x), F_23(x) + F_29(x))
Eq(F_32(x), F_33(x) + F_34(x) + F_35(x))
Eq(F_33(x), 0)
Eq(F_34(x), F_17(x)*F_18(x))
Eq(F_35(x), F_17(x)*F_36(x))
Eq(F_36(x), F_25(x) + F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_17(x)*F_40(x))
Eq(F_40(x), F_26(x) + F_38(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_17(x)*F_43(x))
Eq(F_43(x), F_29(x) + F_41(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.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": "point_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:
{"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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], 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