0123_0132_0213_1203_1230_2013_3012
Counting sequence:
1, 1, 2, 6, 17, 49, 146, 455, 1476, 4941, 16929, 59007, 208376, 743496, 2675412, 9696426, 35360237, 129648953, 477645446, 1767274116, 6564138110, 24466295655, 91482609985, 343059688651, 1289904268692, 4861946597844, 18367353389936, 69533551430205, 263747952582371, 1002242217997607, 3814986504270582, 14544636042751455, 55534064882751052, 212336130421470541, 812944042164661081, 3116285494931459043, 11959798385899541624, 45950804324684988312, 176733862787109035516, 680425371730141380491, 2622127042276760023075, 10113918591638331628415, 39044429911905145367930, 150853479205086486563826, 583300119592998529399898, 2257117854077251044468747, 8740328711533178197573249, 33868773757191054665171491, 131327898242169378064260876, 509552245179617158419619596, 1978261657756160686575054504, 7684785670514316438547107277, 29869166945772626036409988731, 116157871455782434390137708271, 451959718027953471673460943086, 1759414616608818871357915172078, 6852456927844873498140945194134, 26700952856774851905201942938647, 104088460289122304035046327567941, 405944995127576985733148174149013, 1583850964596120042690825518576716, 6182127958584855650493638317536116, 24139737743045626825722068756130972, 94295850558771979787952552626557626, 368479169875816659479036820603582173, 1440418573150919668872534839814078137, 5632681584560312734993988429309393174, 22033725021956517463358670283087410876, 86218923998960285726185831056191817566, 337485502510215975556784101516579794759, 1321422108420282270489942675644241423793, 5175569924646105559418941000510598765671, 20276890389709399862928999873224185954284, 79463489365077377841208239743834346862476, 311496878311103321137536294935263756933872, 1221395654430378811828760727535901831675701, 4790408930363303911328386217339258784815907, 18793142726809884575211361293559879217926183, 73745243611532458459690151878064057587803206, 289450081175264899454283846067379829637536056, 1136359577947336271931632877065973247389225450, 4462290049988320482463241297605328036594410155, 17526585015616776834735140518076156280212601285, 68854441132780194707888052034928342639896328601, 270557451039395118028642463289588762561398770028, 1063353702922273835973036658044156350360742016692, 4180080073556524734514695828172007546207117416164, 16435314834665426797069144960764666122783595109041, 64633260585762914370496637486149061529875906077031, 254224158304000796523953440778846307133157747556819, 1000134600800354781929399250536549404476265836297138, 3935312233584004685417853572763361709934446801900446, 15487357822491889407128326963778362972288150996058674, 60960876535340415751462563580829680832404363897537811, 239993345518077005168915776623476774686989682346552577, 944973797977428207852605870454939680458374248083326931, 3721443204405954385563870541379246795011359042085440252, 14657929356129575437016877846657032980635950785454924028, 57743358069601357782187700608042856688245579804018525976, 227508830794229349661819540395688854529189526615358131341, 896519947090131496687170070074100633348210213731824908395
Generating function in Maple syntax:
1/2*(-(x^2+x-1)*(x-1)^2*(1-4*x)^(1/2)-x^4-x^3-2*x^2+3*x-1)/x/(x^2+x-1)/(x-1)^2
Generating function in latex syntax:
\frac{-\left(x^{2}+x -1\right) \left(x -1\right)^{2} \sqrt{1-4 x}-x^{4}-x^{3}-2 x^{2}+3 x -1}{2 x \left(x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**4/2 - x**3/2 - x**2 + 3*x/2 + sqrt(1 - 4*x)*(x - 1)**2*(-x**2 - x + 1)/2 - 1/2)/(x*(x - 1)**2*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x^2+x-1)^2*(x-1)^4*F(x)^2+(x^2+x-1)*(x^4+x^3+2*x^2-3*x+1)*(x-1)^2*F(x)+x^8-2*x^7-2*x^6+12*x^5-7*x^4-9*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(x^{2}+x -1\right) \left(x^{4}+x^{3}+2 x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-2 x^{7}-2 x^{6}+12 x^{5}-7 x^{4}-9 x^{3}+13 x^{2}-6 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 49
a(6) = 146
a(7) = 455
a(n+5) = 2*(2*n+1)/(6+n)*a(n)+(7*n+10)/(6+n)*a(1+n)-2*(3*n+8)/(6+n)*a(n+2)-(24+7*n)/(6+n)*a(n+3)+2*(3*n+14)/(6+n)*a(n+4)+(3*n^2-10*n-18)/(6+n), n >= 8
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) = 17
a \! \left(5\right) = 49
a \! \left(6\right) = 146
a \! \left(7\right) = 455
a \! \left(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{6+n}+\frac{\left(7 n +10\right) a \! \left(1+n \right)}{6+n}-\frac{2 \left(3 n +8\right) a \! \left(n +2\right)}{6+n}-\frac{\left(24+7 n \right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(3 n +14\right) a \! \left(n +4\right)}{6+n}+\frac{3 n^{2}-10 n -18}{6+n}, \quad n \geq 8
Specification 1
Strategy pack name: requirement_placements_tracked_fusion
Tree: http://permpal.com/tree/22721/
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[33,x]+F[6,x]
F[6,x] = F[15,x]+F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[4,x]
F[10,x] = F[11,x]+F[12,x]
F[11,x] = F[1,x]+F[4,x]
F[12,x] = F[13,x]+F[8,x]
F[13,x] = F[14,x]
F[14,x] = F[4,x]*F[8,x]
F[15,x] = F[16,x]+F[19,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[1,x]+F[16,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[19,x]+F[8,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[16,x]
F[26,x] = F[27,x]
F[27,x] = F[20,x]+F[23,x]+F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[16,x]+F[30,x]
F[30,x] = F[20,x]+F[28,x]+F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[16,x]+F[30,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x,1]
F[35,x,k[0]] = F[36,x,k[0]]
F[36,x,k[0]] = F[37,x,k[0]]
F[37,x,k[0]] = F[38,x,k[0]]*F[4,x]*F[42,x,k[0]]
F[38,x,k[0]] = F[1,x]+F[39,x,k[0]]
F[39,x,k[0]] = F[40,x,k[0]]
F[40,x,k[0]] = F[38,x,k[0]]*F[41,x,k[0]]
F[41,x,k[0]] = k[0]*x
F[42,x,k[0]] = F[43,x,k[0]]+F[52,x,k[0]]
F[43,x,k[0]] = F[44,x,k[0]]+F[47,x,k[0]]
F[44,x,k[0]] = F[16,x]+F[45,x,k[0]]
F[45,x,k[0]] = F[46,x,k[0]]
F[46,x,k[0]] = F[41,x,k[0]]*F[44,x,k[0]]
F[47,x,k[0]] = F[30,x]+F[48,x,k[0]]
F[48,x,k[0]] = 2*F[20,x]+F[49,x,k[0]]+F[51,x,k[0]]
F[49,x,k[0]] = F[4,x]*F[50,x,k[0]]
F[50,x,k[0]] = F[45,x,k[0]]+F[48,x,k[0]]
F[51,x,k[0]] = F[41,x,k[0]]*F[47,x,k[0]]
F[52,x,k[0]] = 1/(-1+k[0])*(F[36,x,k[0]]*k[0]-F[36,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{15}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{12}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x , 1\right)
F_{35}\! \left(x , y\right) = F_{36}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = F_{37}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = F_{38}\! \left(x , y\right) F_{4}\! \left(x \right) F_{42}\! \left(x , y\right)
F_{38}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{39}\! \left(x , y\right)
F_{39}\! \left(x , y\right) = F_{40}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{38}\! \left(x , y\right) F_{41}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = y x
F_{42}\! \left(x , y\right) = F_{43}\! \left(x , y\right)+F_{52}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{44}\! \left(x , y\right)+F_{47}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{16}\! \left(x \right)+F_{45}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{46}\! \left(x , y\right)
F_{46}\! \left(x , y\right) = F_{41}\! \left(x , y\right) F_{44}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{30}\! \left(x \right)+F_{48}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = 2 F_{20}\! \left(x \right)+F_{49}\! \left(x , y\right)+F_{51}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = F_{4}\! \left(x \right) F_{50}\! \left(x , y\right)
F_{50}\! \left(x , y\right) = F_{45}\! \left(x , y\right)+F_{48}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{41}\! \left(x , y\right) F_{47}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = \frac{y F_{36}\! \left(x , y\right)-F_{36}\! \left(x , 1\right)}{-1+y}
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_33(x) + F_6(x))
Eq(F_6(x), F_15(x) + F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_4(x))
Eq(F_10(x), F_11(x) + F_12(x))
Eq(F_11(x), F_1(x) + F_4(x))
Eq(F_12(x), F_13(x) + F_8(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_4(x)*F_8(x))
Eq(F_15(x), F_16(x) + F_19(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_1(x) + F_16(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_19(x) + F_8(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_16(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_20(x) + F_23(x) + F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_16(x) + F_30(x))
Eq(F_30(x), F_20(x) + F_28(x) + F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_16(x) + F_30(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x, 1))
Eq(F_35(x, k_0), F_36(x, k_0))
Eq(F_36(x, k_0), F_37(x, k_0))
Eq(F_37(x, k_0), F_38(x, k_0)*F_4(x)*F_42(x, k_0))
Eq(F_38(x, k_0), F_1(x) + F_39(x, k_0))
Eq(F_39(x, k_0), F_40(x, k_0))
Eq(F_40(x, k_0), F_38(x, k_0)*F_41(x, k_0))
Eq(F_41(x, k_0), k_0*x)
Eq(F_42(x, k_0), F_43(x, k_0) + F_52(x, k_0))
Eq(F_43(x, k_0), F_44(x, k_0) + F_47(x, k_0))
Eq(F_44(x, k_0), F_16(x) + F_45(x, k_0))
Eq(F_45(x, k_0), F_46(x, k_0))
Eq(F_46(x, k_0), F_41(x, k_0)*F_44(x, k_0))
Eq(F_47(x, k_0), F_30(x) + F_48(x, k_0))
Eq(F_48(x, k_0), 2*F_20(x) + F_49(x, k_0) + F_51(x, k_0))
Eq(F_49(x, k_0), F_4(x)*F_50(x, k_0))
Eq(F_50(x, k_0), F_45(x, k_0) + F_48(x, k_0))
Eq(F_51(x, k_0), F_41(x, k_0)*F_47(x, k_0))
Eq(F_52(x, k_0), (k_0*F_36(x, k_0) - F_36(x, 1))/(k_0 - 1))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "limited_insertion": true, "maxreqlen": 2, "strategy_class": "RequirementInsertionFactory"}, {"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.requirement_insertion", "ignore_parent": true, "strategy_class": "RequirementCorroborationFactory"}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "requirement_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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 0, 1, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 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