0132_0231_1320_3012_3102
Counting sequence:
1, 1, 2, 6, 19, 50, 118, 271, 625, 1460, 3452, 8233, 19743, 47494, 114450, 276051, 666141, 1607848, 3881272, 9369741, 22620011, 54608922, 131836910, 318281687, 768399113, 1855078620, 4478554932, 10812186929, 26102927095, 63018039278, 152139003658, 367296044443, 886731090229, 2140758222416, 5168247532400, 12477253284373, 30122754098115, 72722761477378, 175568277049446, 423859315572639, 1023286908190881, 2470433131950340, 5964153172087276, 14398739476120377, 34761632124323279, 83922003724761942, 202605639573841922, 489133282872440291, 1180872205318716749, 2850877693509867768, 6882627592338445992, 16616132878186753181, 40114893348711945499, 96845919575610637034, 233806732499933212126, 564459384575477053543, 1362725501650887311161, 3289910387877251667500, 7942546277405390637476, 19175002942688032933441, 46292552162781456495015, 111760107268250945913790, 269812766699283348312570, 651385640666817642528555, 1572584048032918633358949, 3796553736732654909235360, 9165691521498228451818208, 22127936779729111812859941, 53421565080956452077525875, 128971066941642015967899090, 311363698964240484013311062, 751698464870122983994507823, 1814760628704486452002312913, 4381219722279095887999119444, 10577200073262678228000537180, 25535619868804452344000178761, 61648439810871582916000879231, 148832499490547618176001921318, 359313438791966819268004705522, 867459377074481256712011315571, 2094232192940929332692027319421, 5055923762956339922096065936712, 12206079718853609176884159174680, 29468083200663558275864384267437, 71142246120180725728612927690443, 171752575441025009733090239628730, 414647397002230745194793406927822, 1001047369445486500122677053463799, 2416742135893203745440147513834345, 5834531641231893991002972081110908, 14085805418356991727446091676034068, 34006142477945877445895155433156433, 82098090374248746619236402542323799, 198202323226443370684367960517780366, 478502736827135487987972323577860330, 1155207796880714346660312607673476283, 2788918330588564181308597538924787605, 6733044458057842709277507685523025648, 16255007246704249599863612909970812496, 39243058951466341909004733505464623669, 94741125149636933417873079920900032291
Generating function in Maple syntax:
(x^8+x^7-5*x^6-5*x^5+5*x^2-4*x+1)/(x^2+2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{8}+x^{7}-5 x^{6}-5 x^{5}+5 x^{2}-4 x +1}{\left(x^{2}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**8 + x**7 - 5*x**6 - 5*x**5 + 5*x**2 - 4*x + 1)/((x - 1)**3*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+2*x-1)*(x-1)^3*F(x)+x^8+x^7-5*x^6-5*x^5+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}+x^{7}-5 x^{6}-5 x^{5}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
4+1/2*(2^(1/2)-1)^(-n)+1/2*(-1-2^(1/2))^(-n)-13/2*n+3/2*n^2+piecewise(n = 0,-4,n = 1,1,n = 2,2,n = 3,1,0)
Explicit closed form in latex syntax:
4+\frac{\left(\sqrt{2}-1\right)^{-n}}{2}+\frac{\left(-1-\sqrt{2}\right)^{-n}}{2}-\frac{13 n}{2}+\frac{3 n^{2}}{2}+\left(\left\{\begin{array}{cc}-4 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 50
a(6) = 118
a(7) = 271
a(8) = 625
a(n+2) = -3*n^2+a(n)+2*a(n+1)+13*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) = 50
a \! \left(6\right) = 118
a \! \left(7\right) = 271
a \! \left(8\right) = 625
a \! \left(n +2\right) = -3 n^{2}+a \! \left(n \right)+2 a \! \left(n +1\right)+13 n -5, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19169/
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[26,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[14,x]+F[19,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[4,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[11,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[11,x]*F[4,x]
F[25,x] = F[19,x]
F[26,x] = F[27,x]+F[37,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[27,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[27,x]
F[37,x] = F[38,x]+F[39,x]+F[70,x]
F[38,x] = 0
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[56,x]
F[41,x] = F[42,x]+F[7,x]
F[42,x] = F[38,x]+F[43,x]+F[46,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[11,x]+F[45,x]
F[45,x] = F[43,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[31,x]+F[45,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[4,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[23,x]
F[56,x] = F[37,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[38,x]+F[61,x]+F[68,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[66,x]
F[63,x] = F[11,x]+F[64,x]
F[64,x] = F[38,x]+F[43,x]+F[65,x]
F[65,x] = F[4,x]*F[48,x]
F[66,x] = F[60,x]+F[67,x]
F[67,x] = F[58,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[27,x]+F[60,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[69,x]+F[72,x]
F[72,x] = F[73,x]+F[76,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[31,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[45,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_{26}\! \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_{14}\! \left(x \right)+F_{19}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{37}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{27}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{70}\! \left(x \right)
F_{38}\! \left(x \right) = 0
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{31}\! \left(x \right)+F_{45}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{23}\! \left(x \right)
F_{56}\! \left(x \right) = F_{37}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \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_{38}\! \left(x \right)+F_{61}\! \left(x \right)+F_{68}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{66}\! \left(x \right)
F_{63}\! \left(x \right) = F_{11}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{66}\! \left(x \right) = F_{60}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{58}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{27}\! \left(x \right)+F_{60}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{69}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{31}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{45}\! \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_26(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_14(x) + F_19(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_4(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_11(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_11(x)*F_4(x))
Eq(F_25(x), F_19(x))
Eq(F_26(x), F_27(x) + F_37(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_27(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_27(x))
Eq(F_37(x), F_38(x) + F_39(x) + F_70(x))
Eq(F_38(x), 0)
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_56(x))
Eq(F_41(x), F_42(x) + F_7(x))
Eq(F_42(x), F_38(x) + F_43(x) + F_46(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_11(x) + F_45(x))
Eq(F_45(x), F_43(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_31(x) + F_45(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_4(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_23(x))
Eq(F_56(x), F_37(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_38(x) + F_61(x) + F_68(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_66(x))
Eq(F_63(x), F_11(x) + F_64(x))
Eq(F_64(x), F_38(x) + F_43(x) + F_65(x))
Eq(F_65(x), F_4(x)*F_48(x))
Eq(F_66(x), F_60(x) + F_67(x))
Eq(F_67(x), F_58(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_27(x) + F_60(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_69(x) + F_72(x))
Eq(F_72(x), F_73(x) + F_76(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_31(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_45(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:
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