0123_1032_1302_1320_2031
Counting sequence:
1, 1, 2, 6, 19, 52, 138, 363, 952, 2494, 6531, 17100, 44770, 117211, 306864, 803382, 2103283, 5506468, 14416122, 37741899, 98809576, 258686830, 677250915, 1773065916, 4641946834, 12152774587, 31816376928, 83296356198, 218072691667, 570921718804, 1494692464746, 3913155675435, 10244774561560, 26821168009246, 70218729466179, 183835020389292, 481286331701698, 1260023974715803, 3298785592445712, 8636332802621334, 22610212815418291, 59194305643633540, 154972704115482330, 405723806702813451, 1062198715992958024, 2780872341276060622, 7280418307835223843, 19060382582229610908, 49900729438853608882, 130641805734331215739, 342024687764140038336, 895432257558088899270, 2344272084910126659475, 6137383997172291079156, 16067879906606746577994, 42066255722647948654827, 110130887261337099386488, 288326406061363349504638, 754848330922752949127427, 1976218586706895497877644, 5173807429197933544505506, 13545203700886905135638875, 35461803673462781862411120, 92840207319501440451594486, 243058818285041539492372339, 636336247535623178025522532, 1665949924321827994584195258, 4361513525429860805727063243, 11418590651967754422596994472, 29894258430473402462063920174, 78264184639452452963594766051, 204898295487883956428720377980, 536430701824199416322566367890, 1404393809984714292538978725691, 3676750728129943461294369809184, 9625858374405116091344130701862, 25200824395085404812738022296403, 65976614810851098346869936187348, 172729020037467890227871786265642, 452210445301552572336745422609579, 1183902315867189826782364481563096, 3099496502300016908010348022079710, 8114587191032860897248679584676035, 21244265070798565783735690731948396, 55618208021362836453958392611169154, 145610358993289943578139487101559067, 381212868958506994280460068693508048, 998028247882231039263240718978965078, 2612871874688186123509262088243387187, 6840587376182327331264545545751196484, 17908890253858795870284374549010202266, 46886083385394060279588578101279410315, 122749359902323384968481359754828028680, 321361996321576094625855501163204675726, 841336629062404898909085143734785998499, 2202647890865638602101399930041153319772, 5766607043534510907395114646388673960818, 15097173239737894120083944009124868562683, 39524912675679171452856717380985931727232, 103477564787299620238486208133832926619014, 270907781686219689262601907020512848129811
Generating function in Maple syntax:
(2*x^5-2*x^4-x^3-2*x^2+3*x-1)/(x-1)/(x^2-3*x+1)
Generating function in latex syntax:
\frac{2 x^{5}-2 x^{4}-x^{3}-2 x^{2}+3 x -1}{\left(x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(2*x**5 - 2*x**4 - x**3 - 2*x**2 + 3*x - 1)/((x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^2-3*x+1)*F(x)+2*x^5-2*x^4-x^3-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+2 x^{5}-2 x^{4}-x^{3}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/10*(31*5^(1/2)-65)*(3/2-1/2*5^(1/2))^(-n)-1+1/10*(-31*5^(1/2)-65)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(31 \sqrt{5}-65\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}-1+\frac{\left(-31 \sqrt{5}-65\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10} & \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) = 52
a(n+2) = -a(n)+3*a(n+1)+1, n >= 6
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) = 52
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17929/
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[24,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[7,x]
F[14,x] = F[15,x]+F[16,x]+F[22,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[15,x]+F[16,x]+F[21,x]
F[21,x] = F[4,x]*F[7,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[7,x]
F[24,x] = F[2,x]+F[25,x]
F[25,x] = F[15,x]+F[26,x]+F[65,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[37,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[1,x]+F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[18,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[33,x]
F[37,x] = F[38,x]+F[59,x]
F[38,x] = F[15,x]+F[26,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[15,x]+F[42,x]+F[58,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[36,x]+F[4,x]
F[45,x] = F[41,x]+F[46,x]
F[46,x] = 2*F[15,x]+F[47,x]+F[56,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[4,x]
F[52,x] = F[53,x]
F[53,x] = 2*F[15,x]+F[47,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[41,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[41,x]
F[58,x] = F[2,x]*F[4,x]
F[59,x] = 2*F[15,x]+F[56,x]+F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[50,x]
F[63,x] = F[64,x]
F[64,x] = 2*F[15,x]+F[54,x]+F[60,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[2,x]+F[68,x]
F[68,x] = F[15,x]+F[42,x]+F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[2,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_{24}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{65}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \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_{18}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{59}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{58}\! \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_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)+F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{56}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{47}\! \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_{41}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{41}\! \left(x \right)
F_{58}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{50}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{2}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{2}\! \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_24(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_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_22(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_15(x) + F_16(x) + F_21(x))
Eq(F_21(x), F_4(x)*F_7(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_7(x))
Eq(F_24(x), F_2(x) + F_25(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_65(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_37(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_1(x) + F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_18(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_33(x))
Eq(F_37(x), F_38(x) + F_59(x))
Eq(F_38(x), F_15(x) + F_26(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_58(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_36(x) + F_4(x))
Eq(F_45(x), F_41(x) + F_46(x))
Eq(F_46(x), 2*F_15(x) + F_47(x) + F_56(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_4(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), 2*F_15(x) + F_47(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_41(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_41(x))
Eq(F_58(x), F_2(x)*F_4(x))
Eq(F_59(x), 2*F_15(x) + F_56(x) + F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_50(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), 2*F_15(x) + F_54(x) + F_60(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_2(x) + F_68(x))
Eq(F_68(x), F_15(x) + F_42(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_2(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, 2, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 2, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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