0123_0132_0231_1320_2031

Counting sequence:
1, 1, 2, 6, 19, 56, 155, 414, 1088, 2843, 7421, 19381, 50657, 132491, 346674, 907336, 2375075, 6217554, 16277163, 42613408, 111562416, 292073061, 764655837, 2001893351, 5241022929, 13721173941, 35922497170, 94046315594, 246216447363, 644603023948, 1687592621611, 4418174837666, 11566931887792, 30282620821711, 79280930572909, 207560170892121, 543399582098065, 1422638575396159, 3724516144083938, 9750909856848588, 25528213426454131, 66833730422505446, 174972977841053147, 458085203100644196, 1199282631460868864, 3139762691281951001, 8220005442384971885, 21520253635872951499, 56340755465233868513, 147502012759828638953, 386165282814252032226, 1010993835682927440526, 2646816224234530271027, 6929454837020663353056, 18141548286827459767419, 47495190023461715927206, 124344021783557687990880, 325536875327211348020739, 852266604198076356045213, 2231262937267017720087293, 5841522207602976804187521, 15293303685541912692444531, 40038388849022761273113682, 104821862861526371126862416, 274427199735556352107437699, 718459736345142685195412986, 1880952009299871703478761675, 4924396291554472425240830504, 12892236865363545572243686288, 33752314304536164291490182733, 88364706048244947302226814141, 231341803840198677615190209711, 605660705472351085543343762737, 1585640312576854579014841023901, 4151260232258212651501179251954, 10868140384197783375488696672466, 28453160920335137474964910703395, 74491342376807629049406035373044, 195020866210087749673253195348363, 510571256253455619970353550601898, 1336692902550279110237807456384336, 3499507451397381710743068818475191, 9161829451641866021991398998962317, 23985980903528216355231128178329761, 62796113258942783043701985535941809, 164402358873300132775874828429407271, 430410963360957615283922499752188290, 1126830531209572713075892670827062484, 2950080630267760523943755512728900563, 7723411359593708858755373867359537038, 20220153448513366052322366089349604731, 52937048985946389298211724400689167596, 138590993509325801842312807112717784672, 362835931542031016228726696937464069121, 949916801116767246843867283699674301389, 2486914471808270724302875154161558709651, 6510826614308044926064758178785001697985, 17045565371115864053891399382193446250449, 44625869499039547235609439967795336915138, 116832043126002777652936920521192564352278, 305870259878968785723201321595782355994451

Generating function in Maple syntax:
-(3*x^5-8*x^4+15*x^3-14*x^2+6*x-1)/(x^2-3*x+1)/(x-1)^4

Generating function in latex syntax:
-\frac{3 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}

Generating function in sympy syntax:
(-3*x**5 + 8*x**4 - 15*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^4*F(x)+3*x^5-8*x^4+15*x^3-14*x^2+6*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+3 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1 = 0

Explicit closed form in Maple syntax:
1/30*(-27*5^(1/2)+75)*(3/2-1/2*5^(1/2))^(-n)+1/30*(27*5^(1/2)+75)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-3/2*n^2+10/3*n-4

Explicit closed form in latex syntax:
\frac{\left(-27 \sqrt{5}+75\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(27 \sqrt{5}+75\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-\frac{3 n^{2}}{2}+\frac{10 n}{3}-4

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(n+2) = -a(n)+3*a(n+1)-1/6*n*(n^2-6*n-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) = 56
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{n \left(n^{2}-6 n -1\right)}{6}, \quad n \geq 6

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16137/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]
F[15,x] = F[4,x]
F[16,x] = F[12,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[49,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[24,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[13,x]
F[24,x] = F[18,x]+F[25,x]
F[25,x] = 2*F[19,x]+F[26,x]+F[30,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[23,x]
F[29,x] = F[25,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[44,x]
F[32,x] = F[33,x]
F[33,x] = F[19,x]+F[34,x]+F[43,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[12,x]+F[4,x]
F[37,x] = F[33,x]+F[38,x]
F[38,x] = 2*F[19,x]+F[30,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[12,x]
F[42,x] = F[38,x]
F[43,x] = F[2,x]*F[4,x]
F[44,x] = F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[41,x]+F[48,x]
F[48,x] = F[45,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[2,x]+F[33,x]
F[52,x] = F[45,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[22,x]+F[56,x]
F[56,x] = F[53,x]+F[57,x]
F[57,x] = 2*F[19,x]+F[58,x]+F[61,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[28,x]+F[60,x]
F[60,x] = F[57,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[72,x]
F[63,x] = F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[36,x]+F[67,x]
F[67,x] = F[64,x]+F[68,x]
F[68,x] = 2*F[19,x]+F[61,x]+F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[41,x]+F[71,x]
F[71,x] = F[68,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[41,x]+F[75,x]
F[75,x] = F[73,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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{49}\! \left(x \right)
F_{19}\! \left(x \right) = 0
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_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \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_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{23}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{44}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{43}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{30}\! \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_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \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_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{52}\! \left(x \right) = F_{45}\! \left(x \right)+F_{53}\! \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_{22}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{53}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{19}\! \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_{28}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{57}\! \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_{72}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{36}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{64}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{61}\! \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_{41}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{68}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{41}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{73}\! \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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_12(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_49(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_24(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_13(x))
Eq(F_24(x), F_18(x) + F_25(x))
Eq(F_25(x), 2*F_19(x) + F_26(x) + F_30(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_23(x))
Eq(F_29(x), F_25(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_44(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_19(x) + F_34(x) + F_43(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_12(x) + F_4(x))
Eq(F_37(x), F_33(x) + F_38(x))
Eq(F_38(x), 2*F_19(x) + F_30(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_12(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), F_2(x)*F_4(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_41(x) + F_48(x))
Eq(F_48(x), F_45(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_2(x) + F_33(x))
Eq(F_52(x), F_45(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_22(x) + F_56(x))
Eq(F_56(x), F_53(x) + F_57(x))
Eq(F_57(x), 2*F_19(x) + F_58(x) + F_61(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_28(x) + F_60(x))
Eq(F_60(x), F_57(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_72(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_36(x) + F_67(x))
Eq(F_67(x), F_64(x) + F_68(x))
Eq(F_68(x), 2*F_19(x) + F_61(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_41(x) + F_71(x))
Eq(F_71(x), F_68(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_41(x) + F_75(x))
Eq(F_75(x), F_73(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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