0123_1230_1320_3012_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 112, 214, 378, 638, 1049, 1700, 2735, 4387, 7033, 11283, 18124, 29153, 46954, 75708, 122180, 197316, 318827, 515370, 833313, 1347681, 2179867, 3526289, 5704758, 9229503, 14932564, 24160210, 39090750, 63248762, 102337133, 165583328, 267917699, 433498063, 701412589, 1134907263, 1836316240, 2971219661, 4807531822, 7778747160, 12586274408, 20365016736, 32951286047, 53316297414, 86267577813, 139583869293, 225851440879, 365435303645, 591286737690, 956722034187, 1548008764408, 2504730790798, 4052739547074, 6557470329398, 10610209867649, 17167680187868, 27777890045975, 44945570223931, 72723460259617, 117669030472875, 190392490721428, 308061521182841, 498454011892402, 806515533062964, 1304969544942668, 2111485077992508, 3416454622921619, 5527939700900130, 8944394323807305, 14472334024692537, 23416728348484483, 37889062373161193, 61305790721629374, 99194853094773783, 160500643816385884, 259695496911141898, 420196140727509510, 679891637638632626, 1100087778366122837, 1779979416004735640, 2880067194370838123, 4660046610375552871, 7540113804746369557, 12200160415121900439, 19740274219868247448, 31940434634990124773, 51680708854858348534, 83621143489848449040, 135301852344706772720, 218922995834555196312, 354224848179261942983, 573147844013817112638, 927372692193079028349, 1500520536206896113093, 2427893228399975112919, 3928413764606871196853, 6356306993006846279970
Generating function in Maple syntax:
(x^9-x^8-8*x^7-6*x^6+2*x^5+2*x^4+2*x^3+3*x^2-3*x+1)/(x^2+x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{9}-x^{8}-8 x^{7}-6 x^{6}+2 x^{5}+2 x^{4}+2 x^{3}+3 x^{2}-3 x +1}{\left(x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**9 - x**8 - 8*x**7 - 6*x**6 + 2*x**5 + 2*x**4 + 2*x**3 + 3*x**2 - 3*x + 1)/((x - 1)**3*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(x-1)^3*F(x)+x^9-x^8-8*x^7-6*x^6+2*x^5+2*x^4+2*x^3+3*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}-x^{8}-8 x^{7}-6 x^{6}+2 x^{5}+2 x^{4}+2 x^{3}+3 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
7/2*n^2-piecewise(n = 0,44,n = 1,20,n = 2,5,Or(n = 3,n = 4),-1,0)-65/2*n+37-9/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+9/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+4*(-1/2*5^(1/2)-1/2)^(-n)+4*(1/2*5^(1/2)-1/2)^(-n)
Explicit closed form in latex syntax:
\frac{7 n^{2}}{2}-\left(\left\{\begin{array}{cc}44 & n =0 \\ 20 & n =1 \\ 5 & n =2 \\ -1 & n =3\text{ or } n =4 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)-\frac{65 n}{2}+37-\frac{9 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\frac{9 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+4 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+4 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 112
a(7) = 214
a(8) = 378
a(9) = 638
a(n+2) = -7/2*n^2+a(n)+a(n+1)+79/2*n-59, n >= 10
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) = 51
a \! \left(6\right) = 112
a \! \left(7\right) = 214
a \! \left(8\right) = 378
a \! \left(9\right) = 638
a \! \left(n +2\right) = -\frac{7 n^{2}}{2}+a \! \left(n \right)+a \! \left(n +1\right)+\frac{79 n}{2}-59, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18029/
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[36,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[21,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[14,x]
F[21,x] = F[22,x]+F[23,x]+F[30,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[29,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[11,x]
F[29,x] = F[21,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[11,x]
F[36,x] = F[14,x]+F[37,x]
F[37,x] = F[22,x]+F[38,x]+F[71,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[60,x]
F[40,x] = F[41,x]+F[7,x]
F[41,x] = F[22,x]+F[42,x]+F[49,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[11,x]+F[44,x]
F[44,x] = F[22,x]+F[42,x]+F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[11,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[35,x]*F[4,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[55,x]
F[51,x] = F[11,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[11,x]+F[52,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[48,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[26,x]
F[60,x] = F[37,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,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[54,x]+F[67,x]
F[67,x] = F[64,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[64,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[74,x]
F[73,x] = F[14,x]+F[64,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[33,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[44,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_{36}\! \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_{21}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \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_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{22}\! \left(x \right) = 0
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_{29}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \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_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{22}\! \left(x \right)+F_{38}\! \left(x \right)+F_{71}\! \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_{60}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{7}\! \left(x \right)
F_{41}\! \left(x \right) = F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{49}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \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_{55}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{11}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{48}\! \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_{26}\! \left(x \right)
F_{60}\! \left(x \right) = F_{37}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \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_{54}\! \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) = 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_{64}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{14}\! \left(x \right)+F_{64}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{33}\! \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_{44}\! \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_36(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_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_14(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_30(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_29(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_11(x))
Eq(F_29(x), F_21(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_11(x))
Eq(F_36(x), F_14(x) + F_37(x))
Eq(F_37(x), F_22(x) + F_38(x) + F_71(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_60(x))
Eq(F_40(x), F_41(x) + F_7(x))
Eq(F_41(x), F_22(x) + F_42(x) + F_49(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_11(x) + F_44(x))
Eq(F_44(x), F_22(x) + F_42(x) + F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_11(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_35(x)*F_4(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_55(x))
Eq(F_51(x), F_11(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_11(x) + F_52(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_48(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_26(x))
Eq(F_60(x), F_37(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(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_54(x) + F_67(x))
Eq(F_67(x), F_64(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_64(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_74(x))
Eq(F_73(x), F_14(x) + F_64(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_33(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_44(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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