0132_0231_1032_2031_2301_3120
Counting sequence:
1, 1, 2, 6, 18, 48, 117, 269, 596, 1290, 2751, 5811, 12198, 25496, 53129, 110457, 229224, 474966, 982851, 2031407, 4194074, 8650500, 17825517, 36699861, 75497148, 155188898, 318766727, 654311019, 1342176846, 2751462960, 5637144081, 11542724081, 23622319568, 48318381486, 98784247179, 201863462247, 412316859714, 841813589276, 1717986917621, 3504693312717, 7146825579684, 14568529067130, 29686813949007, 60473139526691, 123145302309878, 250688651131848, 510173395286937, 1037938976619369, 2111062325328696, 4292493394836230, 8725724278029011, 17732923532769951, 36028797018962538, 73183493944769076, 148618787703224829, 301741175033821637, 612489549322385804, 1242993497154255186, 2522015791327475991, 5116089176692881627, 10376293541461620894, 21040817459074955360, 42658095670453336097, 86469112845513521121, 175244068700240738208, 355099823418908866398, 719423018874672510747, 1457292781823054575319, 2951479051793528256146, 5976745079881894721100, 12101064112353465857541, 24497276129886284543421, 49584848070131274741108, 100350287760979960788266, 203061758763394744186079, 410845884009659133588627, 831136500985057557607494, 1681162467901593696072696, 3400103867666144553857961, 6875765599058203431138137, 13902646925568235509117704, 28107525306040128311915190, 56819513521887571211186787, 114847952863389771597083151, 232113757366008801543582138, 469063218010476119785992548, 947797842577869272969638157, 1914938498269572612734578869, 3868562622766813359059759196, 7814496497988962985300717570, 15783735500888598504963829671, 31876956011598542078652444491, 64372882042839774294754455278, 129983704124964928864408039056, 262443288328500618278614330929, 529838336814142757656825163217, 1069580193942568557512843324784, 2158967428513703199424072641806, 4357548938284538567644917263531, 8794326039083341472883378482247, 17747108403195211620953844870114
Generating function in Maple syntax:
-(x^6-2*x^5+5*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{x^{6}-2 x^{5}+5 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**6 + 2*x**5 - 5*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^3*F(x)+x^6-2*x^5+5*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{6}-2 x^{5}+5 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-1/2*n^2-3/2*n+1/8*2^n*n+3*2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -\frac{n^{2}}{2}-\frac{3 n}{2}+\frac{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) = 18
a(5) = 48
a(6) = 117
a(n+2) = -4*a(n)+4*a(n+1)-1/2*(n+2)*(n-3), 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) = 18
a \! \left(5\right) = 48
a \! \left(6\right) = 117
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{\left(n +2\right) \left(n -3\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22376/
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[21,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[21,x]*F[7,x]
F[7,x] = F[68,x]+F[8,x]
F[8,x] = F[47,x]+F[9,x]
F[9,x] = F[10,x]+F[43,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[21,x]
F[13,x] = F[14,x]+F[31,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[21,x]
F[17,x] = F[18,x]+F[22,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[21,x]
F[21,x] = x
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[21,x]*F[25,x]
F[25,x] = F[1,x]+F[24,x]
F[26,x] = F[27,x]+F[28,x]+F[30,x]
F[27,x] = 0
F[28,x] = F[21,x]*F[29,x]
F[29,x] = F[19,x]+F[26,x]
F[30,x] = F[21,x]*F[22,x]
F[31,x] = F[23,x]+F[32,x]
F[32,x] = F[27,x]+F[28,x]+F[33,x]
F[33,x] = F[21,x]*F[34,x]
F[34,x] = F[22,x]+F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]
F[37,x] = F[21,x]*F[38,x]
F[38,x] = F[23,x]+F[36,x]
F[39,x] = 2*F[27,x]+F[40,x]+F[42,x]
F[40,x] = F[21,x]*F[41,x]
F[41,x] = F[26,x]+F[39,x]
F[42,x] = F[21,x]*F[35,x]
F[43,x] = F[11,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[21,x]*F[46,x]
F[46,x] = F[15,x]
F[47,x] = F[48,x]+F[67,x]
F[48,x] = F[21,x]+F[49,x]
F[49,x] = F[27,x]+F[50,x]+F[51,x]
F[50,x] = F[15,x]*F[21,x]
F[51,x] = F[21,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[21,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[19,x]*F[21,x]
F[56,x] = F[57,x]+F[61,x]
F[57,x] = F[27,x]+F[58,x]+F[59,x]
F[58,x] = F[21,x]*F[23,x]
F[59,x] = F[21,x]*F[60,x]
F[60,x] = F[21,x]+F[57,x]
F[61,x] = 2*F[27,x]+F[62,x]+F[63,x]
F[62,x] = F[21,x]*F[26,x]
F[63,x] = F[21,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[55,x]
F[66,x] = 2*F[27,x]+F[62,x]+F[63,x]
F[67,x] = F[49,x]
F[68,x] = F[25,x]*F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[21,x]*F[71,x]
F[71,x] = F[25,x]+F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[21,x]*F[25,x]*F[74,x]
F[74,x] = F[24,x]+F[71,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_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{21}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{68}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{47}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{43}\! \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_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{21}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = x
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{21}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right) F_{22}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{21}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{22}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{26}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{21}\! \left(x \right) F_{35}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{21}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{67}\! \left(x \right)
F_{48}\! \left(x \right) = F_{21}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{27}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right) F_{21}\! \left(x \right)
F_{51}\! \left(x \right) = F_{21}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{21}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{19}\! \left(x \right) F_{21}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{27}\! \left(x \right)+F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right) F_{23}\! \left(x \right)
F_{59}\! \left(x \right) = F_{21}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{21}\! \left(x \right)+F_{57}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{21}\! \left(x \right) F_{26}\! \left(x \right)
F_{63}\! \left(x \right) = F_{21}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{55}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{67}\! \left(x \right) = F_{49}\! \left(x \right)
F_{68}\! \left(x \right) = F_{25}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{21}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{25}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{21}\! \left(x \right) F_{25}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{24}\! \left(x \right)+F_{71}\! \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_21(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_21(x)*F_7(x))
Eq(F_7(x), F_68(x) + F_8(x))
Eq(F_8(x), F_47(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_43(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_21(x))
Eq(F_13(x), F_14(x) + F_31(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_21(x))
Eq(F_17(x), F_18(x) + F_22(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_21(x))
Eq(F_21(x), x)
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_21(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_24(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_30(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_21(x)*F_29(x))
Eq(F_29(x), F_19(x) + F_26(x))
Eq(F_30(x), F_21(x)*F_22(x))
Eq(F_31(x), F_23(x) + F_32(x))
Eq(F_32(x), F_27(x) + F_28(x) + F_33(x))
Eq(F_33(x), F_21(x)*F_34(x))
Eq(F_34(x), F_22(x) + F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_21(x)*F_38(x))
Eq(F_38(x), F_23(x) + F_36(x))
Eq(F_39(x), 2*F_27(x) + F_40(x) + F_42(x))
Eq(F_40(x), F_21(x)*F_41(x))
Eq(F_41(x), F_26(x) + F_39(x))
Eq(F_42(x), F_21(x)*F_35(x))
Eq(F_43(x), F_11(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_21(x)*F_46(x))
Eq(F_46(x), F_15(x))
Eq(F_47(x), F_48(x) + F_67(x))
Eq(F_48(x), F_21(x) + F_49(x))
Eq(F_49(x), F_27(x) + F_50(x) + F_51(x))
Eq(F_50(x), F_15(x)*F_21(x))
Eq(F_51(x), F_21(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_21(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_19(x)*F_21(x))
Eq(F_56(x), F_57(x) + F_61(x))
Eq(F_57(x), F_27(x) + F_58(x) + F_59(x))
Eq(F_58(x), F_21(x)*F_23(x))
Eq(F_59(x), F_21(x)*F_60(x))
Eq(F_60(x), F_21(x) + F_57(x))
Eq(F_61(x), 2*F_27(x) + F_62(x) + F_63(x))
Eq(F_62(x), F_21(x)*F_26(x))
Eq(F_63(x), F_21(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_55(x))
Eq(F_66(x), 2*F_27(x) + F_62(x) + F_63(x))
Eq(F_67(x), F_49(x))
Eq(F_68(x), F_25(x)*F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_21(x)*F_71(x))
Eq(F_71(x), F_25(x) + F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_21(x)*F_25(x)*F_74(x))
Eq(F_74(x), F_24(x) + F_71(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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