0213_0231_1032_3012_3120
Counting sequence:
1, 1, 2, 6, 19, 49, 108, 219, 426, 815, 1556, 2987, 5782, 11287, 22192, 43875, 87090, 173343, 345644, 690011, 1378478, 2755111, 5508040, 11013523, 22024074, 44044719, 88085508, 176166539, 352328006, 704650295, 1409294176, 2818581187, 5637154402, 11274299967, 22548590172, 45097169595, 90194327390, 180388641863, 360777269624, 721554523891, 1443109031098, 2886218044111, 5772436068660, 11544872116203, 23089744209654, 46179488394839, 92358976763408, 184717953498659, 369435906967186, 738871813902175, 1477743627769996, 2955487255503387, 5910974510967822, 11821949021894247, 23643898043744552, 47287796087442515, 94575592174835690, 189151184349619183, 378302368699183204, 756604737398308171, 1513209474796554918, 3026418949593045111, 6052837899186022080, 12105675798371972483, 24211351596743869634, 48422703193487660159, 96845406386975237308, 193690812773950387579, 387381625547900683966, 774763251095801272455, 1549526502191602445016, 3099053004383204785587, 6198106008766409462042, 12396212017532818810127, 24792424035065637501332, 49584848070131274878635, 99169696140262549627990, 198339392280525099121303, 396678784561050198102384, 793357569122100396058851, 1586715138244200791965938, 3173430276488401583774111, 6346860552976803167384300, 12693721105953606334598363, 25387442211907212669020014, 50774884423814425337856679, 101549768847628850675523208, 203099537695257701350849299, 406199075390515402701494346, 812398150781030805402777135, 1624796301562061610805335236, 3249592603124123221610443787, 6499185206248246443220653062, 12998370412496492886441063607, 25996740824992985772881876512, 51993481649985971545763493955, 103986963299971943091526720290, 207973926599943886183053164223, 415947853199887772366106043164, 831895706399775544732211791931, 1663791412799551089464423280158
Generating function in Maple syntax:
(x^8-3*x^7+6*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^{8}-3 x^{7}+6 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**8 - 3*x**7 + 6*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^8-3*x^7+6*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^{8}-3 x^{7}+6 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 = 3,6,2+21/16*2^n+1/3*n^3-n^2-7/3*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 2+\frac{21 \,2^{n}}{16}+\frac{n^{3}}{3}-n^{2}-\frac{7 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) = 49
a(6) = 108
a(7) = 219
a(8) = 426
a(n+1) = -1/3*n^3+2*n^2+2*a(n)+4/3*n-5, 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) = 49
a \! \left(6\right) = 108
a \! \left(7\right) = 219
a \! \left(8\right) = 426
a \! \left(n +1\right) = -\frac{n^{3}}{3}+2 n^{2}+2 a \! \left(n \right)+\frac{4 n}{3}-5, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20339/
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[6,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[4,x]
F[12,x] = F[13,x]+F[16,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[27,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[4,x]
F[26,x] = F[23,x]
F[27,x] = F[28,x]+F[29,x]+F[52,x]
F[28,x] = 0
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[40,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[1,x]+F[32,x]
F[35,x] = F[28,x]+F[36,x]+F[38,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[32,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[14,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[28,x]+F[29,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[18,x]+F[41,x]
F[44,x] = 2*F[28,x]+F[45,x]+F[50,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[49,x]
F[47,x] = F[48,x]
F[48,x] = F[32,x]*F[4,x]
F[49,x] = F[45,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[22,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[17,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[28,x]+F[56,x]+F[62,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[4,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[14,x]*F[4,x]
F[61,x] = F[23,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[21,x]+F[64,x]
F[64,x] = F[55,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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \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_{27}\! \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_{13}\! \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_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{52}\! \left(x \right)
F_{28}\! \left(x \right) = 0
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_{40}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \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_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{28}\! \left(x \right)+F_{45}\! \left(x \right)+F_{50}\! \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_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{22}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{17}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{28}\! \left(x \right)+F_{56}\! \left(x \right)+F_{62}\! \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_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{61}\! \left(x \right) = F_{23}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{21}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{55}\! \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_6(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_4(x))
Eq(F_12(x), F_13(x) + F_16(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_27(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_4(x))
Eq(F_26(x), F_23(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_52(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_40(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_1(x) + F_32(x))
Eq(F_35(x), F_28(x) + F_36(x) + F_38(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_32(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_14(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_28(x) + F_29(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_18(x) + F_41(x))
Eq(F_44(x), 2*F_28(x) + F_45(x) + F_50(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_49(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_32(x)*F_4(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_22(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_17(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_28(x) + F_56(x) + F_62(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_4(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_14(x)*F_4(x))
Eq(F_61(x), F_23(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_21(x) + F_64(x))
Eq(F_64(x), F_55(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, 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, 0, 1, 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, 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, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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