0132_1302_2130_2301_3021
Counting sequence:
1, 1, 2, 6, 19, 52, 126, 280, 587, 1186, 2346, 4594, 8971, 17544, 34430, 67844, 134195, 266278, 529658, 1055438, 2105795, 4205052, 8401822, 16793296, 33573819, 67132042, 134245226, 268467850, 536908827, 1073785936, 2147534686, 4295026044, 8590001891, 17179945934, 34359825530, 68719575334, 137439064595, 274878031748, 549755953598, 1099511783688, 2199023429035, 4398046703602, 8796093235242, 17592186279586, 35184372347819, 70368744462232, 140737488667326, 281474977052020, 562949953794067, 1125899907248886, 2251799814127226, 4503599627850494, 9007199255261411, 18014398510045324, 36028797019572830, 72057594038585024, 144115188076563995, 288230376152473818, 576460752304242538, 1152921504607726138, 2305843009214636475, 4611686018428397152, 9223372036855855262, 18446744073710704876, 36893488147420334019, 73786976294839518622, 147573952589677810426, 295147905179354312790, 590295810358707232307, 1180591620717412982036, 2361183241434824387966, 4722366482869647101944, 9444732965739292427531, 18889465931478582971714, 37778931862957163948330, 75557863725914325784914, 151115727451828649336395, 302231454903657296312488, 604462909807314590132478, 1208925819614629177634788, 2417851639229258352496115, 4835703278458516702069702, 9671406556917033401061882, 19342813113834066798885166, 38685626227668133594364419, 77371252455336267185149212, 154742504910672534366538526, 309485009821345068729130160, 618970019642690137454119547, 1237940039285380274903897386, 2475880078570760549803244906, 4951760157141521099601724394, 9903520314283042199198460251, 19807040628566084398391701104, 39614081257132168796777944030, 79228162514264337593550183004, 158456325028528675187094405795, 316912650057057350374182587758, 633825300114114700748358679418, 1267650600228229401496710581638, 2535301200456458802993414095955
Generating function in Maple syntax:
-(2*x^7-3*x^6+7*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{2 x^{7}-3 x^{6}+7 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-2*x**7 + 3*x**6 - 7*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+2*x^7-3*x^6+7*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{7}-3 x^{6}+7 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,3-19/3*n+23/12*n^2-2/3*n^3+1/12*n^4+2^(n+1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 3-\frac{19 n}{3}+\frac{23 n^{2}}{12}-\frac{2 n^{3}}{3}+\frac{n^{4}}{12}+2^{n +1} & \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(6) = 126
a(7) = 280
a(n+1) = -1/12*n^4+n^3-41/12*n^2+2*a(n)+17/2*n-8, n >= 8
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(6\right) = 126
a \! \left(7\right) = 280
a \! \left(n +1\right) = -\frac{n^{4}}{12}+n^{3}-\frac{41 n^{2}}{12}+2 a \! \left(n \right)+\frac{17 n}{2}-8, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20172/
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[12,x]*F[4,x]
F[4,x] = F[42,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[25,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[19,x]
F[25,x] = F[26,x]+F[27,x]+F[39,x]
F[26,x] = 0
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[10,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[29,x]
F[32,x] = F[25,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[26,x]+F[37,x]+F[39,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[10,x]+F[36,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[21,x]+F[36,x]
F[41,x] = F[34,x]
F[42,x] = F[43,x]+F[51,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[1,x]+F[14,x]
F[47,x] = F[43,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[20,x]
F[51,x] = F[26,x]+F[52,x]+F[71,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[67,x]
F[54,x] = F[55,x]+F[6,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[29,x]+F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[56,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[66,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[10,x]+F[63,x]
F[66,x] = F[61,x]
F[67,x] = F[51,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[35,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[40,x]+F[73,x]
F[73,x] = F[74,x]+F[77,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[14,x]
F[77,x] = F[78,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[59,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{42}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \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_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{25}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{26}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{34}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{51}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)
F_{51}\! \left(x \right) = F_{26}\! \left(x \right)+F_{52}\! \left(x \right)+F_{71}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{67}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{6}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{29}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \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_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{10}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{61}\! \left(x \right)
F_{67}\! \left(x \right) = F_{51}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{35}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{40}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{77}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{14}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{59}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_42(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_25(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_39(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_10(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_29(x))
Eq(F_32(x), F_25(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_26(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_10(x) + F_36(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_21(x) + F_36(x))
Eq(F_41(x), F_34(x))
Eq(F_42(x), F_43(x) + F_51(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_1(x) + F_14(x))
Eq(F_47(x), F_43(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_20(x))
Eq(F_51(x), F_26(x) + F_52(x) + F_71(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_67(x))
Eq(F_54(x), F_55(x) + F_6(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_29(x) + F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_56(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_66(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_10(x) + F_63(x))
Eq(F_66(x), F_61(x))
Eq(F_67(x), F_51(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_35(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_40(x) + F_73(x))
Eq(F_73(x), F_74(x) + F_77(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_14(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_59(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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