0132_0213_1203_2031_3012
Counting sequence:
1, 1, 2, 6, 19, 55, 151, 404, 1069, 2815, 7396, 19408, 50890, 133371, 349412, 915188, 2396697, 6275813, 16432249, 43023414, 112642055, 294909381, 772096882, 2021398806, 5292128004, 13855031365, 36273040886, 94964212434, 248619792559, 650895482755, 1704067169611, 4461306857768, 11679854749585, 30578259568891, 80054927481232, 209586528577228, 548704667477422, 1436527488784863, 3760877823034424, 9846106019405984, 25777440298428885, 67486214978214161, 176681204801793037, 462557399695078506, 1210990994716936139, 3170415585157137825, 8300255761889379646, 21730351702347312114, 56890799348123770824, 148942046346831526345, 389935339700149549226, 1020863972766203389278, 2672656578618825628555, 6997105763123224775311, 18318660710804164987327, 47958876369375537756668, 125557968397462032143797, 328715028823236410107063, 860587118072612633472124, 2253046325395192777037704, 5898551858113922419665490, 15442609248948122490713091, 40429275888732949783254092, 105845218417254779598585356, 277106379363037946482820049, 725473919671869670059730685, 1899315379652588231376547681, 4972472219285922801959945694, 13018101278205225120073500239, 34081831615329825281720801085, 89227393567784368394119361866, 233600349088023470293127991438, 611573653696286350546785780300, 1601120612000836079801241226381, 4191788182306222695372470945822, 10974243934917833311285716537258, 28730943622447279349969756641447, 75218586932424008155078176291163, 196924817174824750643204473114099, 515555864592050252718929566839824, 1349742776601326021985918252078745, 3533672465211927836655553537861171, 9251274619034457525869804734645672, 24220151391891444802259651387684356, 63409179556639876980104002523159734, 166007387278028186298552999998158695, 434612982277444682175250494382435616, 1137831559554305860647394623876634424, 2978881696385472900446825014886076429, 7798813529602112841793168199147693225, 20417558892420865626712658998561713781, 53453863147660484041224875990908260498, 139944030550560586501622015584538594195, 366378228504021275471181284567453864601, 959190654961503239924121998532944872342, 2511193736380488444320924985251248971498, 6574390554179962093070593391855792137872, 17211977926159397834942535899170985760913, 45061543224298231411840635449147013563474, 117972651746735296400714672300614761671094, 308856412015907657790522304448531826614275
Generating function in Maple syntax:
(x^7+x^6+2*x^5-x^4-10*x^3+13*x^2-6*x+1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{7}+x^{6}+2 x^{5}-x^{4}-10 x^{3}+13 x^{2}-6 x +1}{\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**7 + x**6 + 2*x**5 - x**4 - 10*x**3 + 13*x**2 - 6*x + 1)/((x - 1)**3*(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)*(x-1)^3*F(x)-x^7-x^6-2*x^5+x^4+10*x^3-13*x^2+6*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) \left(x -1\right)^{3} F \! \left(x \right)-x^{7}-x^{6}-2 x^{5}+x^{4}+10 x^{3}-13 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/20*(29*5^(1/2)-55)*(3/2-1/2*5^(1/2))^(-n)+1/20*(-29*5^(1/2)-55)*(3/2+1/2*5^(1/2))^(-n)+1/20*(3*5^(1/2)-5)*(-1/2*5^(1/2)-1/2)^(-n)+1/20*(-3*5^(1/2)-5)*(1/2*5^(1/2)-1/2)^(-n)+1/2*n^2-7/2*n+6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(29 \sqrt{5}-55\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(-29 \sqrt{5}-55\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(3 \sqrt{5}-5\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\frac{\left(-3 \sqrt{5}-5\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\frac{n^{2}}{2}-\frac{7 n}{2}+6 & \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) = 55
a(6) = 151
a(7) = 404
a(n+4) = a(n)-2*a(n+1)-3*a(n+2)+4*a(n+3)+1/2*(n-1)*(n-6), n >= 8
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) = 55
a \! \left(6\right) = 151
a \! \left(7\right) = 404
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)+\frac{\left(n -1\right) \left(n -6\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18409/
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[23,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[21,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[4,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[4,x]
F[23,x] = F[24,x]+F[38,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[27,x]*F[4,x]
F[30,x] = F[24,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[37,x]
F[32,x] = 0
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[28,x]
F[36,x] = F[31,x]
F[37,x] = F[30,x]*F[4,x]
F[38,x] = F[32,x]+F[39,x]+F[56,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[45,x]
F[41,x] = F[42,x]+F[7,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[11,x]
F[45,x] = F[38,x]+F[46,x]
F[46,x] = 2*F[32,x]+F[47,x]+F[51,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[42,x]
F[50,x] = F[46,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[24,x]+F[53,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[55,x]+F[58,x]
F[58,x] = F[59,x]+F[68,x]
F[59,x] = F[32,x]+F[33,x]+F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[65,x]
F[62,x] = F[24,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[24,x]*F[4,x]
F[65,x] = F[59,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[59,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[58,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_{23}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
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_{16}\! \left(x \right) F_{4}\! \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_{4}\! \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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{56}\! \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_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)
F_{45}\! \left(x \right) = F_{38}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \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_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{42}\! \left(x \right)
F_{50}\! \left(x \right) = F_{46}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{24}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{68}\! \left(x \right)
F_{59}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = F_{24}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{65}\! \left(x \right) = F_{59}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \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_23(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_21(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_18(x))
Eq(F_17(x), F_1(x) + F_4(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_38(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_27(x)*F_4(x))
Eq(F_30(x), F_24(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_37(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_28(x))
Eq(F_36(x), F_31(x))
Eq(F_37(x), F_30(x)*F_4(x))
Eq(F_38(x), F_32(x) + F_39(x) + F_56(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_45(x))
Eq(F_41(x), F_42(x) + F_7(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_11(x))
Eq(F_45(x), F_38(x) + F_46(x))
Eq(F_46(x), 2*F_32(x) + F_47(x) + F_51(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_42(x))
Eq(F_50(x), F_46(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_24(x) + F_53(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_55(x) + F_58(x))
Eq(F_58(x), F_59(x) + F_68(x))
Eq(F_59(x), F_32(x) + F_33(x) + F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_65(x))
Eq(F_62(x), F_24(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_24(x)*F_4(x))
Eq(F_65(x), F_59(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_59(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_58(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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