0132_1032_1203_1230_2130
Counting sequence:
1, 1, 2, 6, 19, 54, 146, 378, 948, 2318, 5554, 13088, 30418, 69874, 158916, 358326, 801914, 1782832, 3940522, 8664282, 18961796, 41323102, 89710562, 194079296, 418531874, 899920994, 1929772676, 4127830758, 8809066858, 18758591568, 39865357754, 84561525258, 179053498308, 378505398734, 798882371986, 1683657040736, 3543396334834, 7447551467602, 15633807449412, 32779395071382, 68651088419546, 143624242206832, 300167541371722, 626723030649786, 1307321595232004, 2724593486252734, 5673485941754306, 11804366450857088, 24541115089444418, 50982180329255618, 105834631973228804, 219550547736871878, 455145143973713098, 942941344515840144, 1952310717632834906, 4039729317454202346, 8354178039642862404, 17266802153911941422, 35668510962218380402, 73642864202779862432, 151969470835451580946, 313450542170465597362, 646212516447078369732, 1331624359041327972726, 2742800293795049311034, 5647009585120116559024, 11621448856328157943786, 23906980464974499390618, 49160573191774015978628, 101051264416565905170718, 207636551909797755329954, 426488723981072857120448, 875703836280170900413538, 1797450745152197799320354, 3688168227344163067852676, 7565231112390080878154406, 15512973907282663928760874, 31800415913532979516611792, 65168657492545771998357242, 133510745250524066178548682, 273443908899862061200140996, 559883770331636717979020942, 1146061677193058163810965842, 2345316090373381886562715616, 4798226578569854188231057714, 9814059804471951921719716114, 20068168606962627575399746116, 41026106616642231229560992598, 83851094852030638389629795738, 171338638568102298061498544368, 350027546117261365948478434954, 714910372702387336363736932218, 1459840791351685010005035994628, 2980340644619033560259378015614, 6083237353112239540124276636546, 12414062714057154814398134364416, 25328253203946122711415778468994, 51666665479885234857879262104962, 105373456144409414991082564284932, 214866776739393333250049717141382, 438052510542513818242732135575946
Generating function in Maple syntax:
(7*x^6-x^5-7*x^4-5*x^3+12*x^2-6*x+1)/(x^2+x-1)/(2*x-1)^3
Generating function in latex syntax:
\frac{7 x^{6}-x^{5}-7 x^{4}-5 x^{3}+12 x^{2}-6 x +1}{\left(x^{2}+x -1\right) \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(7*x**6 - x**5 - 7*x**4 - 5*x**3 + 12*x**2 - 6*x + 1)/((2*x - 1)**3*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(2*x-1)^3*F(x)+7*x^6-x^5-7*x^4-5*x^3+12*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(2 x -1\right)^{3} F \! \left(x \right)+7 x^{6}-x^{5}-7 x^{4}-5 x^{3}+12 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/160*(-32*5^(1/2)+160)*(-1/2*5^(1/2)-1/2)^(-n)+1/160*(32*5^(1/2)+160)*(1/2*5^(1/2)-1/2)^(-n)+1/32*2^n*(n+14)*(n-3))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-32 \sqrt{5}+160\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{160}+\frac{\left(32 \sqrt{5}+160\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{160}+\\\frac{2^{n} \left(n +14\right) \left(n -3\right)}{32} & \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) = 54
a(6) = 146
a(n+5) = -8*a(n)+4*a(n+1)+14*a(n+2)-17*a(n+3)+7*a(n+4), n >= 7
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) = 54
a \! \left(6\right) = 146
a \! \left(n +5\right) = -8 a \! \left(n \right)+4 a \! \left(n +1\right)+14 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19650/
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[28,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[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]+F[23,x]+F[27,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[10,x]
F[26,x] = F[21,x]
F[27,x] = F[12,x]*F[13,x]
F[28,x] = F[29,x]+F[38,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[1,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[32,x]
F[35,x] = F[29,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[22,x]+F[39,x]+F[52,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[45,x]
F[41,x] = F[10,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[33,x]+F[42,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[22,x]+F[39,x]+F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[29,x]+F[46,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[36,x]+F[49,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[48,x]+F[54,x]
F[54,x] = F[55,x]+F[72,x]
F[55,x] = F[22,x]+F[56,x]+F[66,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[12,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[33,x]
F[61,x] = F[62,x]+F[64,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]*F[29,x]
F[64,x] = F[65,x]
F[65,x] = F[12,x]*F[36,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = F[29,x]+F[62,x]
F[69,x] = F[55,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[12,x]*F[55,x]
F[72,x] = F[22,x]+F[73,x]+F[74,x]+F[78,x]
F[73,x] = 0
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[46,x]
F[77,x] = F[72,x]
F[78,x] = F[12,x]*F[54,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_{28}\! \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_{21}\! \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_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{38}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \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_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{22}\! \left(x \right)+F_{39}\! \left(x \right)+F_{52}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{22}\! \left(x \right)+F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{36}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{72}\! \left(x \right)
F_{55}\! \left(x \right) = F_{22}\! \left(x \right)+F_{56}\! \left(x \right)+F_{66}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{29}\! \left(x \right)+F_{62}\! \left(x \right)
F_{69}\! \left(x \right) = F_{55}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{72}\! \left(x \right) = F_{22}\! \left(x \right)+F_{73}\! \left(x \right)+F_{74}\! \left(x \right)+F_{78}\! \left(x \right)
F_{73}\! \left(x \right) = 0
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{46}\! \left(x \right)
F_{77}\! \left(x \right) = F_{72}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \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_28(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_21(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_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_27(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_10(x))
Eq(F_26(x), F_21(x))
Eq(F_27(x), F_12(x)*F_13(x))
Eq(F_28(x), F_29(x) + F_38(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_1(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_32(x))
Eq(F_35(x), F_29(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_22(x) + F_39(x) + F_52(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_45(x))
Eq(F_41(x), F_10(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_33(x) + F_42(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_22(x) + F_39(x) + F_47(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_29(x) + F_46(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_36(x) + F_49(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_48(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_72(x))
Eq(F_55(x), F_22(x) + F_56(x) + F_66(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_12(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_33(x))
Eq(F_61(x), F_62(x) + F_64(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x)*F_29(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_12(x)*F_36(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), F_29(x) + F_62(x))
Eq(F_69(x), F_55(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_12(x)*F_55(x))
Eq(F_72(x), F_22(x) + F_73(x) + F_74(x) + F_78(x))
Eq(F_73(x), 0)
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_46(x))
Eq(F_77(x), F_72(x))
Eq(F_78(x), F_12(x)*F_54(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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