0132_1320_2130_3021_3120
Counting sequence:
1, 1, 2, 6, 19, 55, 142, 333, 727, 1510, 3036, 5988, 11699, 22793, 44466, 87047, 171119, 337760, 669024, 1328902, 2645243, 5273595, 10524886, 21020785, 42004423, 83961834, 167864836, 335656792, 671224131, 1342339389, 2684547290, 5368936907, 10737685983, 21475149572, 42950037320, 85899768026, 171799178763, 343597943119, 687195407678, 1374390264981, 2748779899447, 5497559079214, 10995117339820, 21990233751564, 43980466454227, 87960931726513, 175921862124930, 351843722761551, 703687443859535, 1407374885864168, 2814749769664944, 5629499537039726, 11258999071543067, 22517998140282851, 45035996277473574, 90071992551542905, 180143985099344807, 360287970194585778, 720575940384677332, 1441151880764440960, 2882303761523518051, 5764607523041189733, 11529215046076016554, 23058430092145117843, 46116860184282730431, 92233720368557326092, 184467440737105846424, 368934881474202172610, 737869762948394064939, 1475739525896777041847, 2951479051793542137998, 5902958103587071420445, 11805916207174129020951, 23611832414348243200630, 47223664828696470479228, 94447329657392923893684, 188894659314785829515251, 377789318629571639483737, 755578637259143258075986, 1511157274518286493842839, 3022314549036572963883055, 6044629098073145902391152, 12089258196146291777753088, 24178516392292583526737622, 48357032784585167022879035, 96714065569170334013242571, 193428131138340667991955318, 386856262276681335947267969, 773712524553362671855678343, 1547425049106725343670178426, 3094850098213450687296748452, 6189700196426901374547345064, 12379400392853802749045877635, 24758800785707605498040160909, 49517601571415210996025820282, 99035203142830421991994102363, 198070406285660843983927496095, 396140812571321687967790974996, 792281625142643375935514481640, 1584563250285286751870957896618, 3169126500570573503741840976459
Generating function in Maple syntax:
-(x^7+2*x^6-16*x^5+30*x^4-33*x^3+21*x^2-7*x+1)/(2*x-1)/(x-1)^6
Generating function in latex syntax:
-\frac{x^{7}+2 x^{6}-16 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1}{\left(2 x -1\right) \left(x -1\right)^{6}}
Generating function in sympy syntax:
(-x**7 - 2*x**6 + 16*x**5 - 30*x**4 + 33*x**3 - 21*x**2 + 7*x - 1)/((x - 1)**6*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^6*F(x)+x^7+2*x^6-16*x^5+30*x^4-33*x^3+21*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{6} F \! \left(x \right)+x^{7}+2 x^{6}-16 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1-47/15*n+1/2*n^2-3/8*n^3+1/120*n^5+5*2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -1-\frac{47 n}{15}+\frac{n^{2}}{2}-\frac{3 n^{3}}{8}+\frac{n^{5}}{120}+5 \,2^{-1+n} & \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) = 55
a(6) = 142
a(7) = 333
a(n+1) = 2*a(n)-1/120*(-1+n)*(n^4-4*n^3-59*n^2+126*n-240), 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) = 19
a \! \left(5\right) = 55
a \! \left(6\right) = 142
a \! \left(7\right) = 333
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(-1+n \right) \left(n^{4}-4 n^{3}-59 n^{2}+126 n -240\right)}{120}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20194/
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[14,x]+F[28,x]
F[28,x] = F[19,x]+F[29,x]+F[46,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[45,x]
F[31,x] = F[32,x]+F[6,x]
F[32,x] = F[19,x]+F[24,x]+F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[10,x]+F[36,x]
F[36,x] = F[19,x]+F[24,x]+F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[24,x]
F[40,x] = 2*F[19,x]+F[41,x]+F[44,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[23,x]+F[43,x]
F[43,x] = F[41,x]
F[44,x] = F[12,x]*F[38,x]
F[45,x] = F[28,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[55,x]
F[48,x] = F[14,x]+F[49,x]
F[49,x] = F[19,x]+F[50,x]+F[54,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[10,x]+F[36,x]
F[53,x] = F[49,x]
F[54,x] = F[12,x]*F[48,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[22,x]
F[59,x] = 2*F[19,x]+F[60,x]+F[66,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[36,x]+F[63,x]
F[63,x] = 2*F[19,x]+F[41,x]+F[64,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[23,x]+F[63,x]
F[66,x] = F[12,x]*F[55,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_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)+F_{46}\! \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_{45}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{6}\! \left(x \right)
F_{32}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{24}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{45}\! \left(x \right) = F_{28}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{55}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{19}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{22}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{60}\! \left(x \right)+F_{66}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{36}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{23}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \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_14(x) + F_28(x))
Eq(F_28(x), F_19(x) + F_29(x) + F_46(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_45(x))
Eq(F_31(x), F_32(x) + F_6(x))
Eq(F_32(x), F_19(x) + F_24(x) + F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_10(x) + F_36(x))
Eq(F_36(x), F_19(x) + F_24(x) + F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_24(x))
Eq(F_40(x), 2*F_19(x) + F_41(x) + F_44(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_23(x) + F_43(x))
Eq(F_43(x), F_41(x))
Eq(F_44(x), F_12(x)*F_38(x))
Eq(F_45(x), F_28(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_55(x))
Eq(F_48(x), F_14(x) + F_49(x))
Eq(F_49(x), F_19(x) + F_50(x) + F_54(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_10(x) + F_36(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_12(x)*F_48(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_22(x))
Eq(F_59(x), 2*F_19(x) + F_60(x) + F_66(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_36(x) + F_63(x))
Eq(F_63(x), 2*F_19(x) + F_41(x) + F_64(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_23(x) + F_63(x))
Eq(F_66(x), F_12(x)*F_55(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, 3, 2, 0], "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]]}, {"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, 3, 2, 0], "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]]}, {"patt": [3, 1, 2, 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