0213_1032_1302_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 56, 157, 431, 1177, 3216, 8800, 24096, 65962, 180401, 492744, 1343926, 3660077, 9953676, 27032447, 73321311, 198633575, 537508816, 1452979922, 3923763796, 10586224244, 28536315921, 76859033422, 206848309046, 556272160167, 1494919448656, 4014753753825, 10775193242911, 28902112700757, 77479153610816, 207588087664644, 555895319235416, 1487875760070942, 3980455421219041, 10643882994288212, 28449647573517126, 76010029271183425, 202996894862930756, 541925049641390467, 1446194982409697711, 3857965688309302819, 10288202726334565536, 27426838631421870262, 73092400946297133036, 194730431401840329832, 518639007063970186721, 1380926864009938436250, 3675822293820703426726, 9781841869796878570427, 26023928163749194178856, 69217315313643782943269, 184055911005582129710031, 489306724696109452410257, 1300505290833366791156656, 3455775924062845765770152, 9180881782381114476676816, 24385480549899474731148050, 64757333946748296946938321, 171933232401819400733324512, 456400722513700871893871126, 1211297301792782130247174869, 3214217923570151298103513276, 8527508324380690186532097927, 22620035875255468749643161151, 59991633922973346104379838047, 159080240932743910201293375536, 421766179370908776370908951450, 1118044193627854253887862494916, 2963327000360619098350556970972, 7852992707550026545361014769521, 20807858223729565937266228011494, 55126147367862963960841684994646, 146025072991092098628937133549647, 386756973534887357282600619720896, 1024215064290759276605079131303017, 2711987967439484357334649087261631, 7180078865656786146418690664085325, 19007138964316057250178878192807136, 50309679004017842757870487244249932, 133148030643131659316432758241953416, 352344469734833242916675768257813894, 932289416394342204316338314158176641, 2466525836137659666955322496320669292, 6524890224254061552435835373489097766, 17258949176641332961087818898098568937, 45646768193484936487148320943977279476, 120714983727241665998583607529393441675, 319204257070709630846961192807713001231, 843982381408867745059149580788715370267, 2231290594845365590542677688079832695936, 5898447932271664466688604289120011789694, 15591181081713462143670107457766490428636, 41207920422955789526642914113969093820112, 108903927637671274789075250695023406381761, 287785079731523072336711489373959337189682, 760423615830775686356628555824546625573446, 2009121463340972181643753541889898388971875
Generating function in Maple syntax:
(x^8-6*x^7+19*x^6-42*x^5+67*x^4-63*x^3+33*x^2-9*x+1)/(x^2-3*x+1)^2/(x-1)^4
Generating function in latex syntax:
\frac{x^{8}-6 x^{7}+19 x^{6}-42 x^{5}+67 x^{4}-63 x^{3}+33 x^{2}-9 x +1}{\left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**8 - 6*x**7 + 19*x**6 - 42*x**5 + 67*x**4 - 63*x**3 + 33*x**2 - 9*x + 1)/((x - 1)**4*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^2*(x-1)^4*F(x)-x^8+6*x^7-19*x^6+42*x^5-67*x^4+63*x^3-33*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)-x^{8}+6 x^{7}-19 x^{6}+42 x^{5}-67 x^{4}+63 x^{3}-33 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/150*((-45*n-48)*5^(1/2)+105*n+150)*(3/2-1/2*5^(1/2))^(-n)+1/150*((45*n+48)*5^(1/2)+105*n+150)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-n^2+11/6*n-2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-45 n -48\right) \sqrt{5}+105 n +150\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{150}+\\\frac{\left(\left(45 n +48\right) \sqrt{5}+105 n +150\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{150}+\frac{n^{3}}{6}-n^{2}+\frac{11 n}{6}-2 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 157
a(7) = 431
a(8) = 1177
a(n+4) = 1/6*n^3-a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)-13/6*n-1, 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) = 56
a \! \left(6\right) = 157
a \! \left(7\right) = 431
a \! \left(8\right) = 1177
a \! \left(n +4\right) = \frac{n^{3}}{6}-a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)-\frac{13 n}{6}-1, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20568/
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[13,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[9,x]
F[13,x] = x
F[14,x] = F[15,x]+F[6,x]
F[15,x] = F[16,x]+F[17,x]+F[33,x]
F[16,x] = 0
F[17,x] = F[13,x]*F[18,x]
F[18,x] = F[19,x]+F[26,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[13,x]*F[22,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[24,x]
F[24,x] = F[13,x]*F[25,x]
F[25,x] = F[19,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[16,x]+F[17,x]+F[28,x]
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[27,x]+F[6,x]
F[30,x] = F[31,x]
F[31,x] = F[13,x]*F[32,x]
F[32,x] = F[26,x]
F[33,x] = F[13,x]*F[34,x]
F[34,x] = F[14,x]+F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[24,x]
F[37,x] = F[38,x]
F[38,x] = F[13,x]*F[39,x]
F[39,x] = F[35,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right)+F_{6}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_14(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_9(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_6(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_33(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_13(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_26(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_13(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_13(x)*F_25(x))
Eq(F_25(x), F_19(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_16(x) + F_17(x) + F_28(x))
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_27(x) + F_6(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_13(x)*F_32(x))
Eq(F_32(x), F_26(x))
Eq(F_33(x), F_13(x)*F_34(x))
Eq(F_34(x), F_14(x) + F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_24(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_13(x)*F_39(x))
Eq(F_39(x), F_35(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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": 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