0213_0231_0312_1203_3120
Counting sequence:
1, 1, 2, 6, 19, 56, 152, 384, 915, 2082, 4572, 9774, 20483, 42308, 86488, 175524, 354443, 713326, 1432364, 2871970, 5753003, 11517216, 23048152, 46112936, 92245859, 184515546, 369059292, 738151734, 1476342195, 2952729372, 5905510712, 11811081164, 23622230683, 47244539238, 94489166828, 188978433514, 377956979483, 755914085176, 1511828311544, 3023656780560, 6047313736243, 12094627666706, 24189255548252, 48378511333566, 96757022928099, 193514046142836, 387028092599832, 774056185543284, 1548112371461675, 3096224743332062, 6192449487108652, 12384898974699954, 24769797949923083, 49539595900412368, 99079191801436568, 198158383603533304, 396316767207777923, 792633534416321226, 1585267068833464924, 3170534137667812550, 6341068275336571283, 12682136550674155596, 25364273101349394552, 50728546202699946396, 101457092405401127739, 202914184810803571926, 405828369621608545772, 811656739243218583034, 1623313478486438751355, 3246626956972879186152, 6493253913945760158392, 12986507827891522210144, 25973015655783046425683, 51946031311566094973698, 103892062623132192191708, 207784125246264386754894, 415568250492528776013763, 831136500985057554669476, 1662273001970115112124504, 3324546003940230227183940, 6649092007880460457458123, 13298184015760920918167886, 26596368031521841839755052, 53192736063043683683103426, 106385472126087367369980779, 212770944252174734743922816, 425541888504349469492001112, 851083777008698938988358984, 1702167554017397877981283235, 3404335108034795755967347642, 6808670216069591511939699932, 13617340432139183023884635734, 27234680864278366047774746483, 54469361728556732095555215228, 108938723457113464191116408248, 217877446914226928382239058284, 435754893828453856764484631003, 871509787656907713528976057926, 1743019575313815427057959202284, 3486039150627630854115925790730, 6972078301255261708231859276763
Generating function in Maple syntax:
(x^6-3*x^5+12*x^4-18*x^3+15*x^2-6*x+1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{x^{6}-3 x^{5}+12 x^{4}-18 x^{3}+15 x^{2}-6 x +1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**6 - 3*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)-x^6+3*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)-x^{6}+3 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1/12*n^4-23/12*n^2-3*n-5+11*2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{n^{4}}{12}-\frac{23 n^{2}}{12}-3 n -5+11 \,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) = 56
a(6) = 152
a(n+1) = 2*a(n)+1/12*n*(-1+n)*(n^2-3*n+14), 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) = 19
a \! \left(5\right) = 56
a \! \left(6\right) = 152
a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{n \left(-1+n \right) \left(n^{2}-3 n +14\right)}{12}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20215/
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[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]+F[24,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[18,x]+F[5,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[6,x]
F[23,x] = F[19,x]
F[24,x] = F[25,x]+F[26,x]+F[62,x]
F[25,x] = 0
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[28,x]+F[36,x]
F[28,x] = F[29,x]+F[6,x]
F[29,x] = F[30,x]
F[30,x] = F[13,x]*F[31,x]
F[31,x] = F[32,x]+F[9,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[13,x]*F[35,x]
F[35,x] = F[10,x]+F[33,x]
F[36,x] = F[24,x]+F[37,x]
F[37,x] = 2*F[25,x]+F[38,x]+F[42,x]
F[38,x] = F[13,x]*F[39,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[29,x]
F[41,x] = F[37,x]
F[42,x] = F[13,x]*F[43,x]
F[43,x] = F[44,x]+F[50,x]
F[44,x] = F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[13,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[10,x]
F[49,x] = F[45,x]
F[50,x] = F[51,x]
F[51,x] = 2*F[25,x]+F[52,x]+F[60,x]
F[52,x] = F[13,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[33,x]
F[55,x] = F[56,x]
F[56,x] = F[13,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[33,x]
F[59,x] = F[56,x]
F[60,x] = F[13,x]*F[61,x]
F[61,x] = F[45,x]+F[51,x]
F[62,x] = F[13,x]*F[63,x]
F[63,x] = F[14,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[25,x]+F[66,x]+F[76,x]
F[66,x] = F[13,x]*F[67,x]
F[67,x] = F[68,x]+F[70,x]
F[68,x] = F[10,x]+F[69,x]
F[69,x] = F[34,x]
F[70,x] = F[45,x]+F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[13,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[69,x]
F[75,x] = F[71,x]
F[76,x] = F[13,x]*F[77,x]
F[77,x] = F[15,x]+F[65,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_{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_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{5}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{6}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{62}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{36}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{6}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{9}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{29}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{50}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{52}\! \left(x \right)+F_{60}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \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_{33}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{13}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{33}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{45}\! \left(x \right)+F_{51}\! \left(x \right)
F_{62}\! \left(x \right) = F_{13}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{14}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{25}\! \left(x \right)+F_{66}\! \left(x \right)+F_{76}\! \left(x \right)
F_{66}\! \left(x \right) = F_{13}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{10}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{34}\! \left(x \right)
F_{70}\! \left(x \right) = F_{45}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{13}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{69}\! \left(x \right)
F_{75}\! \left(x \right) = F_{71}\! \left(x \right)
F_{76}\! \left(x \right) = F_{13}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{15}\! \left(x \right)+F_{65}\! \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))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_24(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_5(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_6(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_62(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_36(x))
Eq(F_28(x), F_29(x) + F_6(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_13(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_9(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_13(x)*F_35(x))
Eq(F_35(x), F_10(x) + F_33(x))
Eq(F_36(x), F_24(x) + F_37(x))
Eq(F_37(x), 2*F_25(x) + F_38(x) + F_42(x))
Eq(F_38(x), F_13(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_29(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), F_13(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_50(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_13(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_10(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), 2*F_25(x) + F_52(x) + F_60(x))
Eq(F_52(x), F_13(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_33(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_13(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_33(x))
Eq(F_59(x), F_56(x))
Eq(F_60(x), F_13(x)*F_61(x))
Eq(F_61(x), F_45(x) + F_51(x))
Eq(F_62(x), F_13(x)*F_63(x))
Eq(F_63(x), F_14(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_25(x) + F_66(x) + F_76(x))
Eq(F_66(x), F_13(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_70(x))
Eq(F_68(x), F_10(x) + F_69(x))
Eq(F_69(x), F_34(x))
Eq(F_70(x), F_45(x) + F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_13(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_69(x))
Eq(F_75(x), F_71(x))
Eq(F_76(x), F_13(x)*F_77(x))
Eq(F_77(x), F_15(x) + F_65(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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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