0123_0213_0231_0312
Counting sequence:
1, 1, 2, 6, 20, 70, 254, 948, 3618, 14058, 55432, 221262, 892346, 3630680, 14885042, 61432382, 255025212, 1064190214, 4461325382, 18780710508, 79357572866, 336466650450, 1431007889744, 6103431668830, 26099839562738, 111877997049648, 480635694869218, 2069101534401846, 8924406284675748, 38561354439558086, 166897502998609742, 723479181352437796, 3140791753917311010, 13653724835676676282, 59432958330544951128, 259021595623269725614, 1130180631217143256874, 4936693587427846014856, 21586198476349817443346, 94481107486455542803310, 413924780228549876818252, 1815038646864621207934470, 7965628849723926716674966, 34987023649614208084478172, 153790963437724327288736514, 676513886260674343952855714, 2978042136972841059276504992, 13118380018106426185990986174, 57824601255192949233852753890, 255044943255463857542407064672, 1125593717071530481081374350786, 4970468699508847446216754449254, 21961106234150643737172047318580, 97083014050863212803237668307270, 429393965229275374327337657433758, 1900133909361515524779573610539156, 8412410052949430399466816816817442, 37261246386113120799420203990292618, 165115626752815314518145810375649064, 731991966115848418881720629023728142, 3246426651318580849586407765162877594, 14403900962931465771073139286081743544, 63932884620294455631449589485716174194, 283878675920482451411734504836838547358, 1260957483762814328184553481561629971292, 5603023649784798697443199876983078550278, 24905436868177722592558895224786802610662, 110741686357082873746856676522368772319052, 492571316969067213898685366847830209099778, 2191613986985140166686753642396963083387890, 9754209529360393322991797803490568039295728, 43425961829123337212845256187121636813904926, 193389367031333638843143935753347714689380306, 861465905446242424899252437496067538664343440, 3838509913045083717484673711866957839247692962, 17108154152704797622918441691100919085655885462, 76270488408420784685422356978182782992664869572, 340110367452828452185310010580803087349351453894, 1517016769334001222126113008084657520086688111342, 6768078133020142791961499279042312001238544082180, 30202453246714748627379298870795789904487095378210, 134808864901505364026011661082109132538502904109658, 601854687883389254236404091405937925541721965471736, 2687568143749716825455377678441364586339738904822126, 12003826854308171718683213897124023428605944354845706, 53625357308703832617456604054419141917442394561026024, 239612145092321952974442490735023427728791155968385234, 1070862206830889785041837829889605563259792264782332622, 4786769831425872931641010568553869193911597015922673260, 21400987148044250884774007607493917246248918363666295814, 95698593536157379846617471514873350646957386956378735414, 428012085517543045619811511839499347688901873040921067964, 1914624158830341550574835960995453416197201334698179836418, 8566164585639456568339345106834238580422890585953321792834, 38332140914852131165463317628748015761161642007952365441856, 171558394654575476817481243497371340152035790861765236795262, 767947639845245198902910910274137437245117908860852107108290, 3438116063615334659232804191527762897568957034989172186525888, 15394916471955550696045961483040914697697650126063708817263490, 68944663378438299659137019374034369234295155539974208165505734, 308808442689689291231222036251695704286595469082138884237508692
Generating function in Maple syntax:
1/2/x*(x^2-x+1-(x^4-2*x^3+7*x^2-6*x+1)^(1/2))
Generating function in latex syntax:
\frac{x^{2}-x +1-\sqrt{x^{4}-2 x^{3}+7 x^{2}-6 x +1}}{2 x}
Generating function in sympy syntax:
(x**2 - x - sqrt(x**4 - 2*x**3 + 7*x**2 - 6*x + 1) + 1)/(2*x)
Implicit equation for the generating function in Maple syntax:
-x*F(x)^2+(x^2-x+1)*F(x)+x-1 = 0
Implicit equation for the generating function in latex syntax:
-x F \! \left(x \right)^{2}+\left(x^{2}-x +1\right) F \! \left(x \right)+x -1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -(n-1)/(5+n)*a(n)+(1+2*n)/(5+n)*a(1+n)-7*(n+2)/(5+n)*a(n+2)+3*(2*n+7)/(5+n)*a(n+3), n >= 4
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(n +4\right) = -\frac{\left(n -1\right) a \! \left(n \right)}{5+n}+\frac{\left(1+2 n \right) a \! \left(1+n \right)}{5+n}-\frac{7 \left(n +2\right) a \! \left(n +2\right)}{5+n}+\frac{3 \left(2 n +7\right) a \! \left(n +3\right)}{5+n}, \quad n \geq 4
Specification 1
Strategy pack name: insertion_point_placements_tracked_fusion
Tree: http://permpal.com/tree/21142/
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[8,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[19,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[11,x]
F[10,x] = F[1,x]+F[8,x]
F[11,x] = F[12,x]+F[6,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[8,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[8,x]
F[17,x] = F[12,x]
F[18,x] = F[6,x]*F[8,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[24,x]*F[8,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[8,x]
F[24,x] = F[25,x]+F[50,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x,1]
F[27,x,k[0]] = F[2,x]+F[28,x,k[0]]
F[28,x,k[0]] = F[29,x,k[0]]
F[29,x,k[0]] = F[30,x,k[0]]*F[36,x,k[0]]
F[30,x,k[0]] = F[27,x,k[0]]+F[31,x,k[0]]
F[31,x,k[0]] = F[32,x,k[0]]+F[37,x,k[0]]
F[32,x,k[0]] = F[33,x,k[0]]*F[6,x]
F[33,x,k[0]] = F[1,x]+F[34,x,k[0]]
F[34,x,k[0]] = F[35,x,k[0]]
F[35,x,k[0]] = F[33,x,k[0]]*F[36,x,k[0]]
F[36,x,k[0]] = k[0]*x
F[37,x,k[0]] = F[19,x]+F[38,x,k[0]]
F[38,x,k[0]] = F[39,x,k[0]]
F[39,x,k[0]] = F[21,x]*F[40,x,k[0]]*F[8,x]
F[40,x,k[0]] = F[41,x,k[0]]+F[46,x,k[0]]
F[41,x,k[0]] = F[42,x,k[0]]+F[45,x,k[0]]
F[42,x,k[0]] = F[43,x,k[0]]
F[43,x,k[0]] = F[36,x,k[0]]*F[44,x,k[0]]
F[44,x,k[0]] = F[22,x]+F[42,x,k[0]]
F[45,x,k[0]] = -k[0]*(F[28,x,1]-F[28,x,k[0]])/(-1+k[0])
F[46,x,k[0]] = F[47,x,k[0]]+F[49,x,k[0]]
F[47,x,k[0]] = F[48,x,k[0]]
F[48,x,k[0]] = F[22,x]*F[34,x,k[0]]*F[6,x]
F[49,x,k[0]] = -k[0]*(F[38,x,1]-F[38,x,k[0]])/(-1+k[0])
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[22,x]*F[6,x]
F[52,x] = F[37,x,1]
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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{6}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{24}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{50}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x , 1\right)
F_{27}\! \left(x , y\right) = F_{2}\! \left(x \right)+F_{28}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = F_{29}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{30}\! \left(x , y\right) F_{36}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{27}\! \left(x , y\right)+F_{31}\! \left(x , y\right)
F_{31}\! \left(x , y\right) = F_{32}\! \left(x , y\right)+F_{37}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{33}\! \left(x , y\right) F_{6}\! \left(x \right)
F_{33}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{34}\! \left(x , y\right)
F_{34}\! \left(x , y\right) = F_{35}\! \left(x , y\right)
F_{35}\! \left(x , y\right) = F_{33}\! \left(x , y\right) F_{36}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = y x
F_{37}\! \left(x , y\right) = F_{19}\! \left(x \right)+F_{38}\! \left(x , y\right)
F_{38}\! \left(x , y\right) = F_{39}\! \left(x , y\right)
F_{39}\! \left(x , y\right) = F_{21}\! \left(x \right) F_{40}\! \left(x , y\right) F_{8}\! \left(x \right)
F_{40}\! \left(x , y\right) = F_{41}\! \left(x , y\right)+F_{46}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = F_{42}\! \left(x , y\right)+F_{45}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{43}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{36}\! \left(x , y\right) F_{44}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{22}\! \left(x \right)+F_{42}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = -\frac{y \left(F_{28}\! \left(x , 1\right)-F_{28}\! \left(x , y\right)\right)}{-1+y}
F_{46}\! \left(x , y\right) = F_{47}\! \left(x , y\right)+F_{49}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{48}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{34}\! \left(x , y\right) F_{6}\! \left(x \right)
F_{49}\! \left(x , y\right) = -\frac{y \left(F_{38}\! \left(x , 1\right)-F_{38}\! \left(x , y\right)\right)}{-1+y}
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{22}\! \left(x \right) F_{6}\! \left(x \right)
F_{52}\! \left(x \right) = F_{37}\! \left(x , 1\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_8(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_19(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_11(x))
Eq(F_10(x), F_1(x) + F_8(x))
Eq(F_11(x), F_12(x) + F_6(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_8(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_8(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_6(x)*F_8(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_24(x)*F_8(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_8(x))
Eq(F_24(x), F_25(x) + F_50(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x, 1))
Eq(F_27(x, k_0), F_2(x) + F_28(x, k_0))
Eq(F_28(x, k_0), F_29(x, k_0))
Eq(F_29(x, k_0), F_30(x, k_0)*F_36(x, k_0))
Eq(F_30(x, k_0), F_27(x, k_0) + F_31(x, k_0))
Eq(F_31(x, k_0), F_32(x, k_0) + F_37(x, k_0))
Eq(F_32(x, k_0), F_33(x, k_0)*F_6(x))
Eq(F_33(x, k_0), F_1(x) + F_34(x, k_0))
Eq(F_34(x, k_0), F_35(x, k_0))
Eq(F_35(x, k_0), F_33(x, k_0)*F_36(x, k_0))
Eq(F_36(x, k_0), k_0*x)
Eq(F_37(x, k_0), F_19(x) + F_38(x, k_0))
Eq(F_38(x, k_0), F_39(x, k_0))
Eq(F_39(x, k_0), F_21(x)*F_40(x, k_0)*F_8(x))
Eq(F_40(x, k_0), F_41(x, k_0) + F_46(x, k_0))
Eq(F_41(x, k_0), F_42(x, k_0) + F_45(x, k_0))
Eq(F_42(x, k_0), F_43(x, k_0))
Eq(F_43(x, k_0), F_36(x, k_0)*F_44(x, k_0))
Eq(F_44(x, k_0), F_22(x) + F_42(x, k_0))
Eq(F_45(x, k_0), -F_28(x, 1) + (k_0*F_28(x, k_0) - F_28(x, 1))/(k_0 - 1))
Eq(F_46(x, k_0), F_47(x, k_0) + F_49(x, k_0))
Eq(F_47(x, k_0), F_48(x, k_0))
Eq(F_48(x, k_0), F_22(x)*F_34(x, k_0)*F_6(x))
Eq(F_49(x, k_0), -F_38(x, 1) + (k_0*F_38(x, k_0) - F_38(x, 1))/(k_0 - 1))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_22(x)*F_6(x))
Eq(F_52(x), F_37(x, 1))
Pack JSON:
{"expansion_strats": [[{"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.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}, {"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_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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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]]}], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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