0132_0231_1302_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 55, 147, 371, 899, 2115, 4867, 11011, 24579, 54275, 118787, 258051, 557059, 1196035, 2555907, 5439491, 11534339, 24379395, 51380227, 108003331, 226492419, 473956355, 989855747, 2063597571, 4294967299, 8925478915, 18522046467, 38386270211, 79456894979, 164282499075, 339302416387, 700079669251, 1443109011459, 2972117368835, 6116033429507, 12575664242691, 25838523252739, 53051436040195, 108851651149827, 223200860438531, 457396837154819, 936783906865155, 1917548278841347, 3923057487904771, 8022036836253699, 16395917393395715, 33495522228568067, 68398419340689411, 139611588448485379, 284852676431183875, 580964351930793987, 1184446701998440451, 2413929400270585859, 4917930793088581635, 10016005571271983107, 20392299112733605891, 41505174165846491139, 84451500212451540995, 171785304186420199427, 349335215895874633731, 710199646837817737219, 1443457723767772413955, 2933032307719818706947, 5958298335808185171971, 12101064112353465860099, 24571063106181122752515, 49879995975310627569667, 101235731476518019268611, 205422942004829566795779, 416748842113246190108675, 845303600433666493251587, 1714219033281681212571651, 3475661731392058877280259, 7045770792441510658834435, 14280436244197807126216707, 28938661807025185869529091, 58632902251309514973249539, 118776961777137316414881795, 240576238103311205766529027, 487197105304695557406588931, 986483468805537406560239619, 1997145454003367396614602755, 4042647940791319960217452547, 8182009947151810254411399171, 16557448025441961176775786499, 33501752313160603689457549315, 67777217150874570050727051267, 137101859350855865445078007811, 277298568799925181577403826179, 560786837796277264529303273475, 1133953075985408331807597789187, 2292664952756524269113178062851, 4634847507084463749222321094659, 9368730217311757920436572127235, 18935530840909176684857004130307, 38267202494389675057681728012291, 77326686613921993491298895527939
Generating function in Maple syntax:
-(x^4+5*x^2-4*x+1)/(x-1)/(2*x-1)^2
Generating function in latex syntax:
-\frac{x^{4}+5 x^{2}-4 x +1}{\left(x -1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(-x**4 - 5*x**2 + 4*x - 1)/((x - 1)*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)^2*F(x)+x^4+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{4}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,3+5/8*2^n*n-3*2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 3+\frac{5 \,2^{n} n}{8}-3 \,2^{-1+n} & \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(n+2) = -4*a(n)+4*a(n+1)+3, n >= 5
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(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+3, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19120/
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[22,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[32,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[9,x]
F[30,x] = F[14,x]
F[31,x] = F[23,x]
F[32,x] = F[18,x]+F[33,x]+F[47,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[46,x]
F[35,x] = F[36,x]+F[6,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[10,x]+F[43,x]
F[46,x] = F[32,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[57,x]
F[49,x] = F[23,x]+F[50,x]
F[50,x] = F[18,x]+F[51,x]+F[56,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[55,x]
F[53,x] = F[10,x]+F[54,x]
F[54,x] = F[37,x]
F[55,x] = F[50,x]
F[56,x] = F[12,x]*F[49,x]
F[57,x] = F[58,x]+F[70,x]
F[58,x] = F[18,x]+F[59,x]+F[68,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[14,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[12,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[12,x]*F[65,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[23,x]+F[58,x]
F[70,x] = F[18,x]+F[71,x]+F[82,x]+F[84,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[43,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[78,x]+F[80,x]
F[78,x] = F[79,x]
F[79,x] = F[10,x]*F[12,x]
F[80,x] = F[81,x]
F[81,x] = F[12,x]*F[77,x]
F[82,x] = F[12,x]*F[83,x]
F[83,x] = F[50,x]+F[70,x]
F[84,x] = F[12,x]*F[57,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_{22}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{47}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{46}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \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_{10}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{32}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{57}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{37}\! \left(x \right)
F_{55}\! \left(x \right) = F_{50}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{70}\! \left(x \right)
F_{58}\! \left(x \right) = F_{18}\! \left(x \right)+F_{59}\! \left(x \right)+F_{68}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{14}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{23}\! \left(x \right)+F_{58}\! \left(x \right)
F_{70}\! \left(x \right) = F_{18}\! \left(x \right)+F_{71}\! \left(x \right)+F_{82}\! \left(x \right)+F_{84}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{43}\! \left(x \right)+F_{74}\! \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_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{80}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{50}\! \left(x \right)+F_{70}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \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_22(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_32(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_9(x))
Eq(F_30(x), F_14(x))
Eq(F_31(x), F_23(x))
Eq(F_32(x), F_18(x) + F_33(x) + F_47(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_46(x))
Eq(F_35(x), F_36(x) + F_6(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_10(x) + F_43(x))
Eq(F_46(x), F_32(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_57(x))
Eq(F_49(x), F_23(x) + F_50(x))
Eq(F_50(x), F_18(x) + F_51(x) + F_56(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_55(x))
Eq(F_53(x), F_10(x) + F_54(x))
Eq(F_54(x), F_37(x))
Eq(F_55(x), F_50(x))
Eq(F_56(x), F_12(x)*F_49(x))
Eq(F_57(x), F_58(x) + F_70(x))
Eq(F_58(x), F_18(x) + F_59(x) + F_68(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_14(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_12(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_12(x)*F_65(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_23(x) + F_58(x))
Eq(F_70(x), F_18(x) + F_71(x) + F_82(x) + F_84(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_43(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_78(x) + F_80(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_10(x)*F_12(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_12(x)*F_77(x))
Eq(F_82(x), F_12(x)*F_83(x))
Eq(F_83(x), F_50(x) + F_70(x))
Eq(F_84(x), F_12(x)*F_57(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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