0231_0312_0321_1320_3102
Counting sequence:
1, 1, 2, 6, 19, 55, 153, 415, 1110, 2944, 7769, 20439, 53671, 140773, 368970, 966658, 2531847, 6630247, 17361101, 45456627, 119014558, 311596396, 815789757, 2135797351, 5591641899, 14639192425, 38326039058, 100339092510, 262691509915, 687735876439, 1800516830049, 4713815763559, 12340932321126, 32308984210168, 84586025180225, 221449099211703, 579761285206927, 1517834777042317, 3973743079305306, 10403394514892122, 27236440552774863, 71305927284854791, 186681341530615637, 488738097677240571, 1279532952100180654, 3349860759592624420, 8770049328246090213, 22960287227683366855, 60110812358910128595, 157372149855690857809, 412005637218912401954, 1078644761818440144054, 2823928648264551783331, 7393141183020752755063, 19355494900871387784105, 50673343519712629448623, 132664535658459400715382, 347320263455977691702512, 909296254709978693550761, 2380568500674775527113367, 6232409247315670045111543, 16316659241274373903707061, 42717568476510913118817642, 111836046188263966201039666, 292790570088290047685403159, 766535664076620839804565415, 2006816422141596196878790493, 5253913602348206138931699075, 13754924384903084333166697150, 36010859552361147361918675804, 94277654272180520367190004109, 246822103264180676855602293799, 646188655520361935930168508411, 1691743863296905819781405819833, 4429042934370356637991103170610, 11595384939814165897615460499918, 30357111885072143972855889356587, 79475950715402270742376375405207, 208070740261134675893698015721841, 544736270068001769299566618458487, 1426138069942870652005275565214598, 3733677939760610219077382749444456, 9774895749338960057588269080938897, 25591009308256270038409943563451511, 66998132175429850194725477077315039, 175403387218033280767572922206472285, 459212029478669992466883639547979898, 1202232701217976697213774780981324170, 3147486074175260100114027837945727455, 8240225521307803604648592651949449799, 21573190489748150716291621171545948389, 56479345947936648548206425835425313419, 147864847354061794934767682361110236366, 387115196114248736266516802247022558228, 1013480740988684413881642931405454845365, 2653327026851804505405692379993956547463, 6946500339566729102379574803626526773667, 18186173991848382801804453013960350319777, 47612021635978419303149345816379362708546, 124649890916086875107830566996377302874982, 326337651112282206020644899312076949508403
Generating function in Maple syntax:
-(2*x^4+3*x^3+x^2-3*x+1)/(x^2-3*x+1)/(x^2+x-1)
Generating function in latex syntax:
-\frac{2 x^{4}+3 x^{3}+x^{2}-3 x +1}{\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right)}
Generating function in sympy syntax:
(-2*x**4 - 3*x**3 - x**2 + 3*x - 1)/((x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^2+x-1)*F(x)+2*x^4+3*x^3+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+2 x^{4}+3 x^{3}+x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-4/5*(3/2-1/2*5^(1/2))^(-n+1)-4/5*(3/2+1/2*5^(1/2))^(-n+1)-4/5*(-1/2*5^(1/2)-1/2)^(-n+1)-4/5*(1/2*5^(1/2)-1/2)^(-n+1)+1/20*(-19*5^(1/2)+59)*(3/2-1/2*5^(1/2))^(-n)+1/20*(19*5^(1/2)+59)*(3/2+1/2*5^(1/2))^(-n)+1/20*(-3*5^(1/2)-13)*(-1/2*5^(1/2)-1/2)^(-n)+1/20*(3*5^(1/2)-13)*(1/2*5^(1/2)-1/2)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{4 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n +1}}{5}-\frac{4 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n +1}}{5}-\frac{4 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n +1}}{5}-\\\frac{4 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n +1}}{5}+\frac{\left(-19 \sqrt{5}+59\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(19 \sqrt{5}+59\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(-3 \sqrt{5}-13\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\\\frac{\left(3 \sqrt{5}-13\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20} & \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+4) = a(n)-2*a(n+1)-3*a(n+2)+4*a(n+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 +4\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20628/
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[13,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[6,x]
F[10,x] = F[11,x]+F[4,x]
F[11,x] = F[12,x]
F[12,x] = F[4,x]*F[7,x]
F[13,x] = F[14,x]+F[26,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[22,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[4,x]
F[22,x] = F[14,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]+F[28,x]+F[41,x]
F[27,x] = 0
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[37,x]
F[30,x] = F[31,x]+F[7,x]
F[31,x] = F[27,x]+F[32,x]+F[35,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[11,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[10,x]
F[37,x] = F[26,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[37,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[13,x]+F[43,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[47,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[11,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{13}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{26}\! \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_{17}\! \left(x \right)+F_{22}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \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_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{41}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{7}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)
F_{37}\! \left(x \right) = F_{26}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{44}\! \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_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{11}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_13(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_6(x))
Eq(F_10(x), F_11(x) + F_4(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_4(x)*F_7(x))
Eq(F_13(x), F_14(x) + F_26(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_22(x))
Eq(F_17(x), F_1(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_17(x) + F_21(x))
Eq(F_21(x), F_4(x))
Eq(F_22(x), F_14(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_41(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_37(x))
Eq(F_30(x), F_31(x) + F_7(x))
Eq(F_31(x), F_27(x) + F_32(x) + F_35(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_11(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_10(x))
Eq(F_37(x), F_26(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_37(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_13(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_47(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_11(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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