0132_0231_0321_2013_3021
Counting sequence:
1, 1, 2, 6, 19, 56, 159, 442, 1211, 3282, 8821, 23558, 62613, 165808, 437873, 1153944, 3036235, 7979324, 20950875, 54971590, 144160487, 377903070, 990335137, 2594675186, 6796836129, 17802124636, 46622120669, 122089403172, 319696420471, 837100521512, 2191806470631, 5738721543538, 15025163466323, 39338379468138, 102993196163533, 269647651473374, 705962643158445, 1848266047805704, 4838887039866185, 12668498151007920, 33166813571587747, 86832354880615700, 227331075703980147, 595162521498766366, 1558159787327202239, 4079323437552606966, 10679823719470151929, 27960174109136915402, 73200751384498727481, 191642185597475533492, 501726016514160405941, 1313536286157470750268, 3438883686383181976783, 9003116461841935443968, 23570469076842344882943, 61708297524084540260554, 161554437006210158010155, 422955040516143697992834, 1107310738585416464414245, 2898977283326496752141750, 7589621327566855905794757, 19869887131719635192810080, 52020040932283178127770657, 136190237394512156100772296, 356550674710017803995087099, 933461793653070283525570796, 2443834720084251101863788939, 6398042394269799132630123382, 16750292518065378517155235991, 43852835270606800861092894222, 114808213515115952950638066001, 300571805717462915759850542498, 786907204522716509866972038993, 2060149809621574044917182529548, 5393542227883780487036809459853, 14120476881113317140497713670484, 36967888429623270383065267192615, 96783188436090692905915959189464, 253381676935317206129118352940055, 663361842483197721070310584759330, 1736703850740949548259556371595267, 4546749710192998106063844470541210, 11903545280744739134642948921057917, 31163886133854608027286946054691726, 81588113124445862406061776767135709, 213600453246736534108586159294952376, 559213246630270849755072251214195449, 1464039286673090234827381694540582112, 3832904613447028294068575032793447251, 10034674553784051526061347804611552452, 26271119048137240041481477182584795811, 68778682591091896113115101346230006478, 180064928726066903327327862062279566703, 471416103588965723927796555252957379878, 1234183382044544088573917944521289945513, 3231134042552094182029669559960307201914, 8459218745626593737986515298658421150825, 22146522194357397592872725462612535231844, 57980347837505020162517359342374342507365, 151794521318276505138450749070900808215660, 397403216117562179740377680883108713990527
Generating function in Maple syntax:
-(3*x^6-2*x^5+3*x^4-11*x^3+13*x^2-6*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{3 x^{6}-2 x^{5}+3 x^{4}-11 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-3*x**6 + 2*x**5 - 3*x**4 + 11*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**2*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^2*F(x)+3*x^6-2*x^5+3*x^4-11*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+3 x^{6}-2 x^{5}+3 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(-48*5^(1/2)+120)*(3/2-1/2*5^(1/2))^(-n)+1/20*(48*5^(1/2)+120)*(3/2+1/2*5^(1/2))^(-n)+n-3/4*2^n-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-48 \sqrt{5}+120\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(48 \sqrt{5}+120\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+n -\frac{3 \,2^{n}}{4}-\\3 & \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) = 159
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+n-3, 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) = 159
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+n -3, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18706/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[17,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[28,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[27,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[22,x]
F[27,x] = F[21,x]*F[4,x]
F[28,x] = F[23,x]+F[29,x]+F[45,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[36,x]
F[31,x] = F[32,x]+F[7,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[11,x]
F[36,x] = F[28,x]+F[37,x]
F[37,x] = 2*F[23,x]+F[38,x]+F[42,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[32,x]
F[41,x] = F[37,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[22,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[62,x]
F[47,x] = F[18,x]+F[48,x]
F[48,x] = F[23,x]+F[49,x]+F[61,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[11,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[35,x]*F[4,x]
F[54,x] = F[48,x]+F[55,x]
F[55,x] = 2*F[23,x]+F[56,x]+F[60,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[52,x]
F[59,x] = F[55,x]
F[60,x] = F[4,x]*F[44,x]
F[61,x] = F[4,x]*F[47,x]
F[62,x] = F[22,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[48,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{17}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)+F_{45}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)
F_{36}\! \left(x \right) = F_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{22}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{62}\! \left(x \right)
F_{47}\! \left(x \right) = F_{18}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{23}\! \left(x \right)+F_{49}\! \left(x \right)+F_{61}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{48}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \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_{52}\! \left(x \right)
F_{59}\! \left(x \right) = F_{55}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{62}\! \left(x \right) = F_{22}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{48}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_17(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_28(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_27(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_22(x))
Eq(F_27(x), F_21(x)*F_4(x))
Eq(F_28(x), F_23(x) + F_29(x) + F_45(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_36(x))
Eq(F_31(x), F_32(x) + F_7(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_11(x))
Eq(F_36(x), F_28(x) + F_37(x))
Eq(F_37(x), 2*F_23(x) + F_38(x) + F_42(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_32(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_22(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_62(x))
Eq(F_47(x), F_18(x) + F_48(x))
Eq(F_48(x), F_23(x) + F_49(x) + F_61(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_11(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_35(x)*F_4(x))
Eq(F_54(x), F_48(x) + F_55(x))
Eq(F_55(x), 2*F_23(x) + F_56(x) + F_60(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_52(x))
Eq(F_59(x), F_55(x))
Eq(F_60(x), F_4(x)*F_44(x))
Eq(F_61(x), F_4(x)*F_47(x))
Eq(F_62(x), F_22(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_48(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, 2, 3, 1], "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, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 2, 3, 1], "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, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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