0213_1032_1302_3120
Counting sequence:
1, 1, 2, 6, 20, 62, 177, 473, 1202, 2940, 6983, 16211, 36960, 83054, 184445, 405645, 884894, 1917104, 4128963, 8847575, 18874604, 40108290, 84934937, 179306801, 377487690, 792723812, 1660944767, 3472884123, 7247757752, 15099494870, 31406948853, 65229816341, 135291470390, 280246616664, 579820585595, 1198295876255, 2473901163204, 5102421148394, 10514079941393, 21646635172665, 44530220925794, 91534343013260, 188016488350647, 385928581350371, 791648371999760, 1622879162598462, 3324923162395757, 6808175999190173, 13933011347178702, 28499341391955200, 58265320179107123, 119063915148608871, 243194379878008220, 496521858917598674, 1013309916158363145, 2067152228963059265, 4215369251218785914, 8592868089022908084, 17509995351216490223, 35668509048774330155, 72634054790231361384, 147862182965828126630, 300912512702387062757, 612201318946235746341, 1245155224975394736230, 2531815624116635961512, 5146641596564964903147, 10459303889793315768623, 21250649172913403464052, 43165381132480350783930, 87658927838267789281793, 177974186823149753993801, 361261035939527858850450, 733147396465512419429084, 1487545442103938242317095, 3017592182553703291554675, 6120186961799060196953024, 12410379116981427621596174, 25160768620729469698575453, 51001558014992168307920045, 103363157577050794437381374, 209446398248234504517848400, 424332962684734840321871267, 859546257746001343216094711, 1740853180245066011576897100, 3525227689996258673443212962, 7137498039004770647465266937, 14449081396034047896088219473, 29246333428117108994491813802, 59189008128332244393614381060, 119770698800860541596490272863, 242326762690113188811503571131, 490224255557010588860053197080, 991589971467589600194198507894, 2005462863642316045336581247445, 4055491568698905780569530962485, 8200114820226358940931798864534, 16578493006109812641449071612664, 33513512743533814802069090997083, 67740078949696008642480077542335, 136906264824648775361643946185764
Generating function in Maple syntax:
-(x^6-2*x^5+7*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{x^{6}-2 x^{5}+7 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**6 + 2*x**5 - 7*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^3*F(x)+x^6-2*x^5+7*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{6}-2 x^{5}+7 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,6+9/8*2^n*n-9*2^(-1+n)+1/2*n^2+3/2*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 6+\frac{9 \,2^{n} n}{8}-9 \,2^{-1+n}+\frac{n^{2}}{2}+\frac{3 n}{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) = 20
a(5) = 62
a(6) = 177
a(n+2) = 1/2*n^2-4*a(n)+4*a(n+1)-1/2*n+3, 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) = 62
a \! \left(6\right) = 177
a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{n}{2}+3, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21380/
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[14,x]*F[4,x]
F[4,x] = F[30,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]*F[18,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[14,x]*F[15,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[10,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]*F[15,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[14,x]*F[20,x]
F[20,x] = F[21,x]+F[23,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[14,x]*F[21,x]
F[23,x] = F[16,x]+F[24,x]
F[24,x] = F[25,x]+F[26,x]+F[29,x]
F[25,x] = 0
F[26,x] = F[14,x]*F[27,x]
F[27,x] = F[24,x]+F[28,x]
F[28,x] = F[22,x]
F[29,x] = F[14,x]*F[23,x]
F[30,x] = F[31,x]+F[42,x]
F[31,x] = F[32,x]
F[32,x] = F[14,x]*F[33,x]
F[33,x] = F[34,x]+F[38,x]
F[34,x] = F[1,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[14,x]*F[15,x]*F[34,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[10,x]*F[21,x]
F[40,x] = F[41,x]
F[41,x] = F[14,x]*F[15,x]*F[28,x]*F[36,x]
F[42,x] = F[43,x]
F[43,x] = F[14,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[20,x]*F[31,x]
F[46,x] = F[47,x]
F[47,x] = F[14,x]*F[21,x]*F[48,x]*F[49,x]
F[48,x] = F[33,x]+F[39,x]
F[49,x] = F[15,x]+F[50,x]
F[50,x] = F[28,x]+F[51,x]
F[51,x] = F[25,x]+F[52,x]+F[54,x]
F[52,x] = F[14,x]*F[53,x]
F[53,x] = F[16,x]+F[51,x]
F[54,x] = F[14,x]*F[50,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{30}\! \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_{14}\! \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_{18}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \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) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{14}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{22}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right) F_{23}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{42}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{34}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{28}\! \left(x \right) F_{36}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right) F_{31}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right) F_{48}\! \left(x \right) F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{28}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{25}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{16}\! \left(x \right)+F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right) F_{50}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_30(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_14(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_18(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_14(x)*F_15(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_10(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)*F_15(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_14(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_23(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_14(x)*F_21(x))
Eq(F_23(x), F_16(x) + F_24(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_29(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_14(x)*F_27(x))
Eq(F_27(x), F_24(x) + F_28(x))
Eq(F_28(x), F_22(x))
Eq(F_29(x), F_14(x)*F_23(x))
Eq(F_30(x), F_31(x) + F_42(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_38(x))
Eq(F_34(x), F_1(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_14(x)*F_15(x)*F_34(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_10(x)*F_21(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_14(x)*F_15(x)*F_28(x)*F_36(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_14(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_20(x)*F_31(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_14(x)*F_21(x)*F_48(x)*F_49(x))
Eq(F_48(x), F_33(x) + F_39(x))
Eq(F_49(x), F_15(x) + F_50(x))
Eq(F_50(x), F_28(x) + F_51(x))
Eq(F_51(x), F_25(x) + F_52(x) + F_54(x))
Eq(F_52(x), F_14(x)*F_53(x))
Eq(F_53(x), F_16(x) + F_51(x))
Eq(F_54(x), F_14(x)*F_50(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": 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, 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": [3, 1, 2, 0], "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": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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