0132_0321_1032_1203_3120
Counting sequence:
1, 1, 2, 6, 19, 51, 120, 266, 569, 1190, 2452, 5002, 10135, 20442, 41106, 82494, 165341, 331118, 662768, 1326178, 2653123, 5307154, 10615374, 21231990, 42465417, 84932486, 169866860, 339735866, 679474159, 1358951050, 2717905162, 5435813742, 10871631285, 21743266782, 43486538216, 86973081554, 173946168731, 347892343618, 695784693958, 1391569395238, 2783138798433, 5566277605494, 11132555220324, 22265110450730, 44530220912327, 89060441836346, 178120883685250, 356241767383966, 712483534782349, 1424967069580110, 2849934139176672, 5699868278370882, 11399736556760435, 22799473113540722, 45598946227102526, 91197892454227414, 182395784908478521, 364791569816982118, 729583139633990748, 1459166279268009498, 2918332558536048543, 5836665117072128234, 11673330234144289274, 23346660468288613070, 46693320936577262437, 93386641873154563006, 186773283746309166040, 373546567492618374066, 747093134985236792139, 1494186269970473630370, 2988372539940947308982, 5976745079881894668422, 11953490159763789389585, 23906980319527578834262, 47813960639055157726036, 95627921278110315512074, 191255842556220631086711, 382511685112441262238618, 765023370224882524545138, 1530046740449765049160958, 3060093480899530098395453, 6120186961799060196867374, 12240373923598120393814224, 24480747847196240787711010, 48961495694392481575507747, 97922991388784963151104466, 195845982777569926302301230, 391691965555139852604698166, 783383931110279705209495529, 1566767862220559410419093830, 3133535724441118820838294092, 6267071448882237641676698362, 12534142897764475283353510735, 25068285795528950566707139402, 50136571591057901133414400746, 100273143182115802266828927534, 200546286364231604533657985301, 401092572728463209067316105118, 802185145456926418134632349128, 1604370290913852836269264841618, 3208740581827705672538529831163
Generating function in Maple syntax:
-(x^9-4*x^8+6*x^7-4*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^{9}-4 x^{8}+6 x^{7}-4 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**9 + 4*x**8 - 6*x**7 + 4*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^9-4*x^8+6*x^7-4*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^{9}-4 x^{8}+6 x^{7}-4 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,n = 3,6,n = 4,19,5-47/6*n+n^2-1/6*n^3+81/32*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 19 & n =4 \\ 5-\frac{47 n}{6}+n^{2}-\frac{n^{3}}{6}+\frac{81 \,2^{n}}{32} & \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) = 51
a(6) = 120
a(7) = 266
a(8) = 569
a(9) = 1190
a(n+1) = 1/6*n^3-3/2*n^2+2*a(n)+28/3*n-12, n >= 10
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) = 51
a \! \left(6\right) = 120
a \! \left(7\right) = 266
a \! \left(8\right) = 569
a \! \left(9\right) = 1190
a \! \left(n +1\right) = \frac{n^{3}}{6}-\frac{3 n^{2}}{2}+2 a \! \left(n \right)+\frac{28 n}{3}-12, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19358/
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[36,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[10,x]
F[34,x] = F[35,x]
F[35,x] = F[31,x]
F[36,x] = F[15,x]+F[37,x]+F[52,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[10,x]+F[43,x]
F[43,x] = F[41,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[15,x]+F[37,x]+F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[20,x]+F[45,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[35,x]+F[51,x]
F[51,x] = F[49,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[47,x]+F[54,x]
F[54,x] = F[55,x]+F[65,x]
F[55,x] = F[15,x]+F[56,x]+F[63,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[62,x]
F[58,x] = F[10,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[10,x]
F[62,x] = F[35,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[20,x]+F[35,x]
F[65,x] = F[15,x]+F[66,x]+F[67,x]+F[69,x]
F[66,x] = 0
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[45,x]
F[69,x] = F[12,x]*F[54,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_{36}\! \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_{10}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{52}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{20}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{47}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{65}\! \left(x \right)
F_{55}\! \left(x \right) = F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{63}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{10}\! \left(x \right)
F_{62}\! \left(x \right) = F_{35}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{20}\! \left(x \right)+F_{35}\! \left(x \right)
F_{65}\! \left(x \right) = F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = 0
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{45}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \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_36(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_10(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_15(x) + F_37(x) + F_52(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_10(x) + F_43(x))
Eq(F_43(x), F_41(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_15(x) + F_37(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_20(x) + F_45(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_35(x) + F_51(x))
Eq(F_51(x), F_49(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_47(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_65(x))
Eq(F_55(x), F_15(x) + F_56(x) + F_63(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_62(x))
Eq(F_58(x), F_10(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_10(x))
Eq(F_62(x), F_35(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_20(x) + F_35(x))
Eq(F_65(x), F_15(x) + F_66(x) + F_67(x) + F_69(x))
Eq(F_66(x), 0)
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_45(x))
Eq(F_69(x), F_12(x)*F_54(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, 0, 3, 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, 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, 0, 3, 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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