0132_0213_0231_1032_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 138, 329, 752, 1676, 3678, 7991, 17240, 36994, 79030, 168181, 356672, 754072, 1589758, 3342963, 7013112, 14680974, 30671926, 63964401, 133170624, 276825764, 574621598, 1191184559, 2466253272, 5100276506, 10536094838, 21743275501, 44828725120, 92341801264, 190052307710, 390842029291, 803158890232, 1649267448102, 3384434236278, 6940667157993, 14224931692672, 29137058145084, 59648505816606, 122045790693671, 249589139516504, 510173395300274, 1042337023144758, 2128654511388389, 4345269952985792, 8866461766401736, 18084767253676798, 36873221949114211, 75153818781764600, 153122387330617534, 311874274195428790, 635007547459263201, 1292533093055356992, 2630102182384395732, 5350276357316176798, 10880696699727147423, 22121681369643907032, 44963938679667064394, 91369029240092656886, 185620362241702398941, 377005332006438998784, 765539879058946431584, 1554138188210029765118, 3154393236604333369819, 6401020193577214456312, 12986507827891524385366, 26341950537257239757558, 53421770837462861532121, 108319281200822487143680, 219590041453438502493804, 445083041010464061450270, 901971998228102235877911, 1827555828870552697764952, 3702335322569801847604962, 7499117974796996599419318, 15187130608908779007319253, 30752050536447129631664192, 62259679710153402497446904, 126030516694825091463200766, 255083347938686755863088211, 516211324975446657599625464, 1044511908147039606946227694, 2113202332686371797386490678, 4274761698157328761761136849, 8646237461883827857498672832, 17485903054905996382950235396, 35358662372088674101806345118, 71491037268730710875424537231, 144529499586568147094472870360, 292153849271349744876193438074, 590497398739126391126882380150, 1193374197871106585002755881421, 2411507196527920775503494122112, 4872531994627256762002953083792, 9844099192397343945997835971838, 19886268791080348735979531681483, 40168678394732019159926782972152
Generating function in Maple syntax:
-(x^8-3*x^7-2*x^6+9*x^5-20*x^4+28*x^3-20*x^2+7*x-1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
-\frac{x^{8}-3 x^{7}-2 x^{6}+9 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-x**8 + 3*x**7 + 2*x**6 - 9*x**5 + 20*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^4*F(x)+x^8-3*x^7-2*x^6+9*x^5-20*x^4+28*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+x^{8}-3 x^{7}-2 x^{6}+9 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/48*(15*n+21)*2^n+1/6*n^3-3/2*n^2+4/3*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(15 n +21\right) 2^{n}}{48}+\frac{n^{3}}{6}-\frac{3 n^{2}}{2}+\frac{4 n}{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) = 54
a(6) = 138
a(7) = 329
a(8) = 752
a(n+2) = 1/6*n^3-5/2*n^2-4*a(n)+4*a(n+1)+22/3*n-2, n >= 9
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) = 54
a \! \left(6\right) = 138
a \! \left(7\right) = 329
a \! \left(8\right) = 752
a \! \left(n +2\right) = \frac{n^{3}}{6}-\frac{5 n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{22 n}{3}-2, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21566/
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[8,x]
F[4,x] = F[35,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[21,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[8,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[8,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[29,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[8,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[23,x]+F[24,x]+F[27,x]
F[27,x] = F[28,x]*F[8,x]
F[28,x] = F[18,x]+F[26,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[28,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[8,x]
F[34,x] = F[18,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[8,x]
F[37,x] = F[38,x]+F[51,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[20,x]*F[41,x]*F[8,x]
F[41,x] = F[42,x]+F[48,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[1,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[20,x]*F[43,x]*F[8,x]
F[46,x] = F[18,x]*F[47,x]
F[47,x] = F[1,x]+F[8,x]
F[48,x] = F[49,x]
F[49,x] = F[50,x]*F[8,x]
F[50,x] = F[11,x]+F[22,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]*F[58,x]
F[53,x] = F[41,x]+F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[43,x]*F[8,x]
F[56,x] = F[57,x]
F[57,x] = F[18,x]*F[8,x]
F[58,x] = F[59,x]
F[59,x] = F[20,x]*F[60,x]*F[8,x]
F[60,x] = F[1,x]+F[61,x]
F[61,x] = F[58,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{35}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \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_{29}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{8}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \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_{40}\! \left(x \right) = F_{20}\! \left(x \right) F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{1}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right) F_{43}\! \left(x \right) F_{8}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right) F_{8}\! \left(x \right)
F_{50}\! \left(x \right) = F_{11}\! \left(x \right)+F_{22}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right) F_{58}\! \left(x \right)
F_{53}\! \left(x \right) = F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{43}\! \left(x \right) F_{8}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{18}\! \left(x \right) F_{8}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{20}\! \left(x \right) F_{60}\! \left(x \right) F_{8}\! \left(x \right)
F_{60}\! \left(x \right) = F_{1}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{58}\! \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_8(x))
Eq(F_4(x), F_35(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_21(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_8(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_8(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_29(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_8(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_23(x) + F_24(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_8(x))
Eq(F_28(x), F_18(x) + F_26(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_28(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_8(x))
Eq(F_34(x), F_18(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_8(x))
Eq(F_37(x), F_38(x) + F_51(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_20(x)*F_41(x)*F_8(x))
Eq(F_41(x), F_42(x) + F_48(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_1(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_20(x)*F_43(x)*F_8(x))
Eq(F_46(x), F_18(x)*F_47(x))
Eq(F_47(x), F_1(x) + F_8(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_50(x)*F_8(x))
Eq(F_50(x), F_11(x) + F_22(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x)*F_58(x))
Eq(F_53(x), F_41(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_43(x)*F_8(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_18(x)*F_8(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_20(x)*F_60(x)*F_8(x))
Eq(F_60(x), F_1(x) + F_61(x))
Eq(F_61(x), F_58(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "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, 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": [1, 0, 3, 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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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