0213_0321_2301_3120
Counting sequence:
1, 1, 2, 6, 20, 58, 146, 336, 732, 1542, 3182, 6484, 13112, 26394, 52986, 106200, 212660, 425614, 851558, 1703484, 3407376, 6815202, 13630898, 27262336, 54525260, 109051158, 218103006, 436206756, 872414312, 1744829482, 3489659882, 6979320744, 13958642532, 27917286174, 55834573526, 111669148300, 223338297920, 446676597234, 893353195938, 1786706393424, 3573412788476, 7146825578662, 14293651159118, 28587302320116, 57174604642200, 114349209286458, 228698418575066, 457396837152376, 914793674307092, 1829587348616622, 3659174697235782, 7318349394474204, 14636698788951152, 29273397577905154, 58546795155813266, 117093590311629600, 234187180623262380, 468374361246528054, 936748722493059518, 1873497444986122564, 3746994889972248776, 7493989779944501322, 14987979559889006538, 29975959119778017096, 59951918239556038340, 119903836479112080958, 239807672958224166326, 479615345916448337196, 959230691832896679072, 1918461383665793362962, 3836922767331586730882, 7673845534663173466864, 15347691069326346938972, 30695382138652693883334, 61390764277305387772206, 122781528554610775550100, 245563057109221551106040, 491126114218443102218074, 982252228436886204442298, 1964504456873772408890904, 3929008913747544817788276, 7858017827495089635583182, 15716035654990179271173158, 31432071309980358542353276, 62864142619960717084713680, 125728285239921434169434658, 251456570479842868338876786, 502913140959685736677761216, 1005826281919371473355530252, 2011652563838742946711068502, 4023305127677485893422145182, 8046610255354971786844298724, 16093220510709943573688605992, 32186441021419887147377220714, 64372882042839774294754450346, 128745764085679548589508909800, 257491528171359097179017828900, 514983056342718194358035667294, 1029966112685436388716071344278, 2059932225370872777432142698444, 4119864450741745554864285406976
Generating function in Maple syntax:
-(2*x^6-3*x^4+2*x^3-6*x^2+4*x-1)/(2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{2 x^{6}-3 x^{4}+2 x^{3}-6 x^{2}+4 x -1}{\left(2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-2*x**6 + 3*x**4 - 2*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^3*F(x)+2*x^6-3*x^4+2*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+2 x^{6}-3 x^{4}+2 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,4-5*n-n^2+13/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 4-5 n -n^{2}+\frac{13 \,2^{n}}{4} & \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) = 58
a(6) = 146
a(n+1) = 2*a(n)+(n+5)*(-2+n), 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) = 58
a \! \left(6\right) = 146
a \! \left(n +1\right) = 2 a \! \left(n \right)+\left(n +5\right) \left(-2+n \right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21372/
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[15,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[15,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[15,x]*F[16,x]
F[11,x] = F[12,x]+F[19,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]*F[16,x]
F[15,x] = x
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[15,x]*F[16,x]
F[19,x] = F[20,x]
F[20,x] = F[15,x]*F[16,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]
F[23,x] = F[15,x]*F[24,x]
F[24,x] = F[25,x]+F[52,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[17,x]+F[5,x]
F[27,x] = F[28,x]
F[28,x] = F[15,x]*F[29,x]
F[29,x] = F[30,x]+F[48,x]
F[30,x] = F[31,x]+F[45,x]
F[31,x] = F[32,x]+F[44,x]
F[32,x] = F[16,x]*F[21,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[1,x]+F[15,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]
F[37,x] = F[15,x]*F[38,x]
F[38,x] = F[1,x]+F[37,x]
F[39,x] = F[40,x]+F[41,x]+F[43,x]
F[40,x] = 0
F[41,x] = F[15,x]*F[42,x]
F[42,x] = F[15,x]+F[39,x]
F[43,x] = F[15,x]*F[36,x]
F[44,x] = F[13,x]*F[16,x]*F[38,x]
F[45,x] = F[38,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[15,x]*F[17,x]*F[19,x]*F[38,x]
F[48,x] = F[38,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[13,x]*F[16,x]
F[51,x] = F[17,x]*F[19,x]
F[52,x] = F[53,x]
F[53,x] = F[15,x]*F[54,x]
F[54,x] = F[55,x]+F[68,x]
F[55,x] = F[56,x]+F[65,x]
F[56,x] = F[57,x]+F[64,x]
F[57,x] = F[12,x]*F[16,x]*F[58,x]
F[58,x] = F[34,x]+F[59,x]
F[59,x] = F[36,x]+F[60,x]
F[60,x] = F[40,x]+F[61,x]+F[63,x]
F[61,x] = F[15,x]*F[62,x]
F[62,x] = F[15,x]+F[60,x]
F[63,x] = F[15,x]*F[36,x]
F[64,x] = F[16,x]*F[19,x]*F[38,x]
F[65,x] = F[38,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[13,x]*F[15,x]*F[17,x]*F[38,x]
F[68,x] = F[38,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[16,x]*F[19,x]
F[71,x] = F[13,x]*F[17,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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \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_{15}\! \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_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{19}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{52}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{5}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{48}\! \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_{44}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = 0
F_{41}\! \left(x \right) = F_{15}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right) F_{36}\! \left(x \right)
F_{44}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{38}\! \left(x \right)
F_{45}\! \left(x \right) = F_{38}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right) F_{38}\! \left(x \right)
F_{48}\! \left(x \right) = F_{38}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{51}\! \left(x \right) = F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{68}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{64}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{34}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{36}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{40}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{15}\! \left(x \right) F_{36}\! \left(x \right)
F_{64}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{38}\! \left(x \right)
F_{65}\! \left(x \right) = F_{38}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{38}\! \left(x \right)
F_{68}\! \left(x \right) = F_{38}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{71}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_22(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_15(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_15(x)*F_16(x))
Eq(F_11(x), F_12(x) + F_19(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x)*F_16(x))
Eq(F_15(x), x)
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_15(x)*F_16(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_15(x)*F_16(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_15(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_52(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_17(x) + F_5(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_15(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_48(x))
Eq(F_30(x), F_31(x) + F_45(x))
Eq(F_31(x), F_32(x) + F_44(x))
Eq(F_32(x), F_16(x)*F_21(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_1(x) + F_15(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_15(x)*F_38(x))
Eq(F_38(x), F_1(x) + F_37(x))
Eq(F_39(x), F_40(x) + F_41(x) + F_43(x))
Eq(F_40(x), 0)
Eq(F_41(x), F_15(x)*F_42(x))
Eq(F_42(x), F_15(x) + F_39(x))
Eq(F_43(x), F_15(x)*F_36(x))
Eq(F_44(x), F_13(x)*F_16(x)*F_38(x))
Eq(F_45(x), F_38(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_15(x)*F_17(x)*F_19(x)*F_38(x))
Eq(F_48(x), F_38(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_13(x)*F_16(x))
Eq(F_51(x), F_17(x)*F_19(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_15(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_68(x))
Eq(F_55(x), F_56(x) + F_65(x))
Eq(F_56(x), F_57(x) + F_64(x))
Eq(F_57(x), F_12(x)*F_16(x)*F_58(x))
Eq(F_58(x), F_34(x) + F_59(x))
Eq(F_59(x), F_36(x) + F_60(x))
Eq(F_60(x), F_40(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_15(x)*F_62(x))
Eq(F_62(x), F_15(x) + F_60(x))
Eq(F_63(x), F_15(x)*F_36(x))
Eq(F_64(x), F_16(x)*F_19(x)*F_38(x))
Eq(F_65(x), F_38(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_13(x)*F_15(x)*F_17(x)*F_38(x))
Eq(F_68(x), F_38(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_16(x)*F_19(x))
Eq(F_71(x), F_13(x)*F_17(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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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