0123_0132_1302_2103_2130
Counting sequence:
1, 1, 2, 6, 19, 49, 120, 298, 732, 1788, 4350, 10556, 25572, 61878, 149616, 361576, 873522, 2109840, 5095176, 12303386, 29707116, 71725980, 173172606, 418093084, 1009394196, 2436938790, 5883364512, 14203817864, 34291243026, 82786696752, 199865272152, 482518269514, 1164903475260, 2812327912572, 6789563657022, 16391462275772, 39572499614340, 95536479959382, 230645489393808, 556827507062632, 1344300581695410, 3245428796945424, 7835158380254568, 18915745888614842, 45666650693312844, 110249048142229404, 266164748380589118, 642578547173213980, 1551321846399640884, 3745222245914925894, 9041766347844546624, 21828754957161503240, 52699276287340091154, 127227307572571707696, 307153891498386066744, 741535090675976423530, 1790224073022874056348, 4321983237000892261116, 10434190547476361446014, 25190364332684485745468, 60814919214027906396708, 146820202762653742590966, 354455324742431409090480, 855730852252526022335848, 2065917029255588932837938, 4987564910776818828651408, 12041046850830447009856200, 29069658612472048208718938, 70180364075830099207364652, 169430386764222137763873948, 409041137604419821655608830, 987512661973297119136013596, 2384066461551394844909054292, 5755645585076702931996462438, 13895357631705797616925737696, 33546360848489911196914036616, 80988079328688229949843668242, 195522519505870594066757329200, 472033118340436250992604140056, 1139588756186754151931367378826, 2751210630713962443643986480636, 6642010017614707983887389692540, 16035230665943425244875462801086, 38712471349501634251763061582524, 93460173364946816359983029189316, 225632818079395465361435309472150, 544725809523738068084141280867792, 1315084437126872120920711693452904, 3174894683777483150317846122752946, 7664873804681839781339679216183312, 18504642293141164913172761287323432, 44674158390964173167644033800258554, 107852959075069517008595217629472780, 260380076541103216504927689810264732, 628613112157275965098678206742695102, 1517606300855655171102604933539408412, 3663825713868586346784436513557958260, 8845257728592827928552347230635524742, 21354341171054242307250548684545653888, 51553940070701312710295731579423678472, 124462221312456867998445716532806502930
Generating function in Maple syntax:
(x+1)*(x^8+3*x^7-2*x^6-5*x^5+4*x^4+x^3+2*x^2-3*x+1)/(x^2+x-1)/(x^2+2*x-1)
Generating function in latex syntax:
\frac{\left(x +1\right) \left(x^{8}+3 x^{7}-2 x^{6}-5 x^{5}+4 x^{4}+x^{3}+2 x^{2}-3 x +1\right)}{\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(x + 1)*(x**8 + 3*x**7 - 2*x**6 - 5*x**5 + 4*x**4 + x**3 + 2*x**2 - 3*x + 1)/((x**2 + x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(x^2+2*x-1)*F(x)+(x+1)*(x^8+3*x^7-2*x^6-5*x^5+4*x^4+x^3+2*x^2-3*x+1) = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+\left(x +1\right) \left(x^{8}+3 x^{7}-2 x^{6}-5 x^{5}+4 x^{4}+x^{3}+2 x^{2}-3 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,13,n = 1,-5,n = 2,2,n = 3,-2,Or(n = 4,n = 5),1,0)+1/5*(-5^(1/2)-5)*(-1/2*5^(1/2)-1/2)^(-n)+1/5*(5^(1/2)-5)*(1/2*5^(1/2)-1/2)^(-n)+1/5*(-20*2^(1/2)-25)*(-1-2^(1/2))^(-n)+1/5*(20*2^(1/2)-25)*(2^(1/2)-1)^(-n)
Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}13 & n =0 \\ -5 & n =1 \\ 2 & n =2 \\ -2 & n =3 \\ 1 & n =4\text{ or } n =5 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{\left(-\sqrt{5}-5\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+\frac{\left(\sqrt{5}-5\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+\frac{\left(-20 \sqrt{2}-25\right) \left(-1-\sqrt{2}\right)^{-n}}{5}+\frac{\left(20 \sqrt{2}-25\right) \left(\sqrt{2}-1\right)^{-n}}{5}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 49
a(6) = 120
a(7) = 298
a(8) = 732
a(9) = 1788
a(n+1) = -1/3*a(n)+a(n+3)-1/3*a(n+4), n >= 10
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) = 49
a \! \left(6\right) = 120
a \! \left(7\right) = 298
a \! \left(8\right) = 732
a \! \left(9\right) = 1788
a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{3}+a \! \left(n +3\right)-\frac{a \! \left(n +4\right)}{3}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16582/
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[17,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[16,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[4,x]
F[16,x] = F[4,x]*F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[13,x]+F[19,x]+F[35,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[13,x]+F[14,x]+F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[26,x]+F[34,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[31,x]
F[29,x] = F[30,x]+F[7,x]
F[30,x] = F[16,x]
F[31,x] = F[26,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[26,x]*F[4,x]
F[34,x] = F[33,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[43,x]
F[37,x] = F[2,x]+F[38,x]
F[38,x] = F[13,x]+F[39,x]+F[42,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]+F[4,x]
F[42,x] = F[2,x]*F[4,x]
F[43,x] = F[44,x]+F[52,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[49,x]
F[47,x] = F[2,x]+F[48,x]
F[48,x] = F[42,x]
F[49,x] = F[44,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[44,x]
F[52,x] = F[51,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_{17}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{35}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{34}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{31}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{26}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{33}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \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_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{52}\! \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_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{2}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{42}\! \left(x \right)
F_{49}\! \left(x \right) = F_{44}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{52}\! \left(x \right) = F_{51}\! \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_17(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_16(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_4(x)*F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_13(x) + F_19(x) + F_35(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_13(x) + F_14(x) + F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_34(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_31(x))
Eq(F_29(x), F_30(x) + F_7(x))
Eq(F_30(x), F_16(x))
Eq(F_31(x), F_26(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_26(x)*F_4(x))
Eq(F_34(x), F_33(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_43(x))
Eq(F_37(x), F_2(x) + F_38(x))
Eq(F_38(x), F_13(x) + F_39(x) + F_42(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x) + F_4(x))
Eq(F_42(x), F_2(x)*F_4(x))
Eq(F_43(x), F_44(x) + F_52(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) + F_49(x))
Eq(F_47(x), F_2(x) + F_48(x))
Eq(F_48(x), F_42(x))
Eq(F_49(x), F_44(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_44(x))
Eq(F_52(x), F_51(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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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