0132_1032_2130_3120
Counting sequence:
1, 1, 2, 6, 20, 62, 177, 475, 1220, 3036, 7379, 17609, 41406, 96178, 221093, 503703, 1138568, 2555768, 5701479, 12648277, 27918146, 61341486, 134217497, 292552451, 635436780, 1375731412, 2969566907, 6392118945, 13723762310, 29393682026, 62813896269, 133949292079, 285078453776, 605590388208, 1284195220943, 2718714297773, 5746666241418, 12128987643238, 25563645345089, 53807350283547, 113112258706676, 237494511598796, 498078767381667, 1043436534758521, 2183630092762190, 4565172278524962, 9534964836072437, 19896762416233415, 41482374692731800, 86412817850170216, 179862510118107959, 374080244048460549, 777433885674830546, 1614540466412321438, 3350678122763647593, 6949054225032673843, 14402511608330844668, 29831843931702163908, 61753358090504239499, 127758114229246228817, 264163139743043811606, 545908332431342041306, 1127557231505496340637, 2327748517801224042591, 4803070988192124500000, 9905901567582029215712, 20420545689596473636767, 42077023232131487233885, 86662803658287473489690, 178416908680918783227606, 367163994043114915362449, 755283489353963881363019, 1553068277053754569651716, 3192319742419880164455868, 6559367044705937201821043, 13472911575627097794671913, 27663622856650381661828318, 56781734590024614049477778, 116510225865359886712304709, 238989522965067004974724087, 490068304126280301696512936, 1004617356099756844180829016, 2058800670803713484524614407, 4217942184635441190549844661, 8638983906966140082450330210, 17689002592601254084300763662, 36209746149097373040799379897, 74102317039098309892789760355, 151608969186231415541552115980, 310103979840987758862230614196, 634134785123936045817076379739, 1296432706141614492888107839489, 2649810654090356478421575397286, 5414749731834253322408769351498, 11062232191054158136498574061293, 22594881597036760777458815331983, 46140501144244693606040158072368, 94202285229460297398723756944848, 192286750422119547339531167460719, 392417088933151264100823186009357, 800679810369155395720355162090666
Generating function in Maple syntax:
(5*x^6-16*x^5+35*x^4-42*x^3+26*x^2-8*x+1)/(2*x-1)^3/(x-1)^3
Generating function in latex syntax:
\frac{5 x^{6}-16 x^{5}+35 x^{4}-42 x^{3}+26 x^{2}-8 x +1}{\left(2 x -1\right)^{3} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(5*x**6 - 16*x**5 + 35*x**4 - 42*x**3 + 26*x**2 - 8*x + 1)/((x - 1)**3*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^3*(x-1)^3*F(x)-5*x^6+16*x^5-35*x^4+42*x^3-26*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{3} \left(x -1\right)^{3} F \! \left(x \right)-5 x^{6}+16 x^{5}-35 x^{4}+42 x^{3}-26 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/16*(n^2+n+6)*2^n-1/2*n^2+1/2*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(n^{2}+n +6\right) 2^{n}}{16}-\frac{n^{2}}{2}+\frac{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+3) = 8*a(n)-12*a(n+1)+6*a(n+2)+1/2*(n-1)*(n-6), 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 +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+\frac{\left(n -1\right) \left(n -6\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15538/
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[12,x]*F[4,x]
F[4,x] = F[27,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]+F[20,x]+F[26,x]
F[19,x] = 0
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[10,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[22,x]
F[25,x] = F[18,x]
F[26,x] = F[12,x]*F[13,x]
F[27,x] = F[28,x]+F[33,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[1,x]+F[14,x]
F[32,x] = F[28,x]
F[33,x] = F[19,x]+F[34,x]+F[41,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[10,x]+F[18,x]
F[37,x] = F[38,x]
F[38,x] = F[19,x]+F[34,x]+F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[28,x]+F[38,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[40,x]+F[43,x]
F[43,x] = F[44,x]+F[56,x]
F[44,x] = F[19,x]+F[45,x]+F[52,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[10,x]+F[48,x]
F[48,x] = F[19,x]+F[49,x]+F[51,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[22,x]
F[51,x] = F[12,x]*F[17,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[28,x]
F[55,x] = F[44,x]
F[56,x] = F[19,x]+F[57,x]+F[67,x]+F[71,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[23,x]+F[60,x]
F[60,x] = 2*F[19,x]+F[61,x]+F[66,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[23,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[12,x]*F[63,x]
F[66,x] = F[12,x]*F[25,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[69,x]+F[70,x]
F[69,x] = F[38,x]
F[70,x] = F[56,x]
F[71,x] = F[12,x]*F[43,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{27}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{26}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{12}\! \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_{10}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right)
F_{33}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \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_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{28}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{56}\! \left(x \right)
F_{44}\! \left(x \right) = F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{22}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{28}\! \left(x \right)
F_{55}\! \left(x \right) = F_{44}\! \left(x \right)
F_{56}\! \left(x \right) = F_{19}\! \left(x \right)+F_{57}\! \left(x \right)+F_{67}\! \left(x \right)+F_{71}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{23}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{61}\! \left(x \right)+F_{66}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{23}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{38}\! \left(x \right)
F_{70}\! \left(x \right) = F_{56}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_27(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_26(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_10(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_22(x))
Eq(F_25(x), F_18(x))
Eq(F_26(x), F_12(x)*F_13(x))
Eq(F_27(x), F_28(x) + F_33(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_1(x) + F_14(x))
Eq(F_32(x), F_28(x))
Eq(F_33(x), F_19(x) + F_34(x) + F_41(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_10(x) + F_18(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_19(x) + F_34(x) + F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_28(x) + F_38(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_40(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_56(x))
Eq(F_44(x), F_19(x) + F_45(x) + F_52(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_10(x) + F_48(x))
Eq(F_48(x), F_19(x) + F_49(x) + F_51(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_22(x))
Eq(F_51(x), F_12(x)*F_17(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_28(x))
Eq(F_55(x), F_44(x))
Eq(F_56(x), F_19(x) + F_57(x) + F_67(x) + F_71(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_23(x) + F_60(x))
Eq(F_60(x), 2*F_19(x) + F_61(x) + F_66(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_23(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_12(x)*F_63(x))
Eq(F_66(x), F_12(x)*F_25(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_70(x))
Eq(F_69(x), F_38(x))
Eq(F_70(x), F_56(x))
Eq(F_71(x), F_12(x)*F_43(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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