0132_0321_1203_1302_3120
Counting sequence:
1, 1, 2, 6, 19, 52, 126, 284, 613, 1288, 2660, 5432, 11011, 22212, 44666, 89636, 179649, 359760, 720080, 1440832, 2882463, 5765868, 11532838, 23066956, 46135389, 92272472, 184546876, 369095944, 738194363, 1476391508, 2952786130, 5905575732, 11811155321, 23622314912, 47244634536, 94489274256, 188978554199, 377957114620, 755914236030, 1511828479452, 3023656966933, 6047313942568, 12094627894548, 24189255799256, 48378511609459, 96757023230692, 193514046474026, 387028092961604, 774056185937713, 1548112371890928, 3096224743798400, 6192449487614432, 12384898975247631, 24769797950515212, 49539595901051606, 99079191802125676, 198158383604275149, 396316767208575480, 792633534417177580, 1585267068834383272, 3170534137668796203, 6341068275337623668, 12682136550675280258, 25364273101350595156, 50728546202701226729, 101457092405402491712, 202914184810805023576, 405828369621610089264, 811656739243220222663, 1623313478486440491548, 3246626956972881031470, 6493253913945762113532, 12986507827891524279941, 25973015655783048615112, 51946031311566097287876, 103892062623132194635896, 207784125246264389334499, 415568250492528778734340, 831136500985057557536730, 1662273001970115115144292, 3324546003940230230362273, 6649092007880460460801168, 13298184015760920921681968, 26596368031521841843446656, 53192736063043683686979199, 106385472126087367374047532, 212770944252174734748187526, 425541888504349469496470924, 851083777008698938993041213, 1702167554017397877986185368, 3404335108034795755972477340, 6808670216069591511945065032, 13617340432139183023890244251, 27234680864278366047780606612, 54469361728556732095561335346, 108938723457113464191122796916, 217877446914226928382245724249, 435754893828453856764491583200, 871509787656907713528983305480, 1743019575313815427057966754512, 3486039150627630854115933657143
Generating function in Maple syntax:
-(2*x^7-3*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{2 x^{7}-3 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:
(-2*x**7 + 3*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)+2*x^7-3*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)+2 x^{7}-3 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+11/4*2^n-59/6*n+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}}{4}-\frac{59 n}{6}+n^{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) = 52
a(6) = 126
a(7) = 284
a(n+1) = 2*a(n)+1/6*(-2+n)*(n^2-7*n+54), 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) = 52
a \! \left(6\right) = 126
a \! \left(7\right) = 284
a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(-2+n \right) \left(n^{2}-7 n +54\right)}{6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19414/
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[19,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[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[20,x]+F[35,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[9,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[16,x]
F[29,x] = F[20,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[24,x]
F[34,x] = F[30,x]
F[35,x] = F[15,x]+F[36,x]+F[40,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[6,x]
F[39,x] = F[35,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[49,x]
F[42,x] = F[20,x]+F[43,x]
F[43,x] = F[15,x]+F[44,x]+F[48,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[10,x]
F[47,x] = F[43,x]
F[48,x] = F[12,x]*F[42,x]
F[49,x] = F[50,x]+F[57,x]
F[50,x] = F[15,x]+F[51,x]+F[55,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[46,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[51,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[20,x]+F[54,x]
F[57,x] = F[15,x]+F[58,x]+F[59,x]+F[61,x]
F[58,x] = 0
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[43,x]
F[61,x] = F[12,x]*F[49,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_{19}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{35}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{9}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{49}\! \left(x \right)
F_{42}\! \left(x \right) = F_{20}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{57}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{20}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = 0
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{43}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \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_19(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_20(x) + F_35(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_9(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_16(x))
Eq(F_29(x), F_20(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_24(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_15(x) + F_36(x) + F_40(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_6(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_49(x))
Eq(F_42(x), F_20(x) + F_43(x))
Eq(F_43(x), F_15(x) + F_44(x) + F_48(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_10(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_12(x)*F_42(x))
Eq(F_49(x), F_50(x) + F_57(x))
Eq(F_50(x), F_15(x) + F_51(x) + F_55(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_46(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_51(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_20(x) + F_54(x))
Eq(F_57(x), F_15(x) + F_58(x) + F_59(x) + F_61(x))
Eq(F_58(x), 0)
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_43(x))
Eq(F_61(x), F_12(x)*F_49(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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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