0213_0321_1032_1230_3120
Counting sequence:
1, 1, 2, 6, 19, 48, 105, 213, 417, 806, 1557, 3023, 5909, 11624, 22985, 45625, 90809, 181066, 361453, 722083, 1443181, 2885196, 5769025, 11536461, 23071089, 46140078, 92277765, 184552823, 369102597, 738201776, 1476399737, 2952795233, 5905585769, 11811166354, 23622327005, 47244647755, 94489288669, 188978569876, 377957131633, 755914254453, 1511828499361, 3023656988406, 6047313965685, 12094627919391, 24189255825909, 48378511638008, 96757023261225, 193514046506633, 387028092996377, 774056185974746, 1548112371930317, 3096224743840243, 6192449487658829, 12384898975294684, 24769797950565025, 49539595901104285, 99079191802181329, 198158383604333886, 396316767208637413, 792633534417242823, 1585267068834451941, 3170534137668868416, 6341068275337699545, 12682136550675359921, 25364273101350678729, 50728546202701314338, 101457092405402583485, 202914184810805119643, 405828369621610189757, 811656739243220327716, 1623313478486440601297, 3246626956972881146053, 6493253913945762233089, 12986507827891524404614, 25973015655783048745045, 51946031311566097423215, 103892062623132194776789, 207784125246264389481096, 415568250492528778886793, 831136500985057557695193, 1662273001970115115308921, 3324546003940230230533226, 6649092007880460460978605, 13298184015760920921866051, 26596368031521841843637549, 53192736063043683687177068, 106385472126087367374252545, 212770944252174734748399853, 425541888504349469496690737, 851083777008698938993268686, 1702167554017397877986420677, 3404335108034795755972720663, 6808670216069591511945316549, 13617340432139183023890504144, 27234680864278366047780875065, 54469361728556732095561612545, 108938723457113464191123083049, 217877446914226928382246019506, 435754893828453856764491887773, 871509787656907713528983619563, 1743019575313815427057967078301
Generating function in Maple syntax:
-(x^7+3*x^6-7*x^5+4*x^4-8*x^3+10*x^2-5*x+1)/(2*x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{x^{7}+3 x^{6}-7 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-x**7 - 3*x**6 + 7*x**5 - 4*x**4 + 8*x**3 - 10*x**2 + 5*x - 1)/((x - 1)**4*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^4*F(x)+x^7+3*x^6-7*x^5+4*x^4-8*x^3+10*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{7}+3 x^{6}-7 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,9+11/8*2^n-23/3*n+1/2*n^2+1/6*n^3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 9+\frac{11 \,2^{n}}{8}-\frac{23 n}{3}+\frac{n^{2}}{2}+\frac{n^{3}}{6} & \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) = 48
a(6) = 105
a(7) = 213
a(n+1) = -1/6*n^3+2*a(n)+55/6*n-16, 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) = 48
a \! \left(6\right) = 105
a \! \left(7\right) = 213
a \! \left(n +1\right) = -\frac{n^{3}}{6}+2 a \! \left(n \right)+\frac{55 n}{6}-16, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20492/
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[17,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[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = x
F[13,x] = F[1,x]+F[12,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[9,x]
F[17,x] = F[18,x]+F[31,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[18,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[12,x]
F[29,x] = F[30,x]
F[30,x] = F[26,x]
F[31,x] = F[32,x]+F[33,x]+F[58,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[45,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[1,x]+F[36,x]
F[39,x] = F[32,x]+F[40,x]+F[43,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[36,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[12,x]+F[42,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[32,x]+F[33,x]+F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[18,x]+F[46,x]
F[49,x] = 2*F[32,x]+F[50,x]+F[55,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[36,x]
F[54,x] = F[50,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[30,x]+F[57,x]
F[57,x] = F[55,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[17,x]+F[60,x]
F[60,x] = F[61,x]+F[70,x]
F[61,x] = F[32,x]+F[62,x]+F[68,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[67,x]
F[64,x] = F[12,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = x^2
F[67,x] = F[30,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[18,x]+F[30,x]
F[70,x] = F[71,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[60,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_{17}\! \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_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \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) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \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_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{58}\! \left(x \right)
F_{32}\! \left(x \right) = 0
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_{45}\! \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_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{32}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{36}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{18}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{50}\! \left(x \right)+F_{55}\! \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_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{54}\! \left(x \right) = F_{50}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{30}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{17}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{70}\! \left(x \right)
F_{61}\! \left(x \right) = F_{32}\! \left(x \right)+F_{62}\! \left(x \right)+F_{68}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{67}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = x^{2}
F_{67}\! \left(x \right) = F_{30}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{60}\! \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_17(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_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + F_12(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_9(x))
Eq(F_17(x), F_18(x) + F_31(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_18(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_12(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_26(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_58(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_45(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_1(x) + F_36(x))
Eq(F_39(x), F_32(x) + F_40(x) + F_43(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_36(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_12(x) + F_42(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_32(x) + F_33(x) + F_47(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_18(x) + F_46(x))
Eq(F_49(x), 2*F_32(x) + F_50(x) + F_55(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_36(x))
Eq(F_54(x), F_50(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_30(x) + F_57(x))
Eq(F_57(x), F_55(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_17(x) + F_60(x))
Eq(F_60(x), F_61(x) + F_70(x))
Eq(F_61(x), F_32(x) + F_62(x) + F_68(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_67(x))
Eq(F_64(x), F_12(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), x**2)
Eq(F_67(x), F_30(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_18(x) + F_30(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_60(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, 3, 2, 1], "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, 2, 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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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