0231_0312_0321_1302_1320_2301_3021
Counting sequence:
1, 1, 2, 6, 17, 47, 127, 339, 898, 2368, 6227, 16347, 42869, 112349, 294322, 770850, 2018605, 5285575, 13839107, 36233343, 94863506, 248361356, 650227327, 1702331571, 4456785097, 11668052377, 30547418402, 79974277854, 209375536553, 548152528223, 1435082365927, 3757095083787, 9836203717474, 25751517414904, 67418350705547, 176503538226315, 462092269676285, 1209773280030005, 3167227585344082, 8291909500160058, 21708500954224261, 56833593425758711, 148792279425386027, 389543245015979511, 1019837455890466802, 2669969123088915332, 6990069914077687927, 18300240620279051619, 47910651948595778833, 125431715228479499953, 328384493741650248002, 859721766004249986102, 2250780804283685979329, 5892620646867172962959, 15427081136350784189647, 40388622762238495897155, 105738787150450971073090, 276827738689254001184560, 724744428917536883914307, 1897405548063722085854523, 4967472215274220660379141, 13005011097759896617308941, 34047561078007017200303602, 89137672136263659714383826, 233365455330788014682385757, 610958693856106941803093287, 1599510626237543420936751827, 4187573184856540488687339759, 10963208928332105823015302738, 28702053600139821925928781308, 75142951872087432678231289327, 196726802016122593777795547667, 515037454176280539047646062809, 1348385560512719331426663810889, 3530119227361877953686357249122, 9241972121572915336147940985870, 24195797137356869359727010637145, 63345419290497694854518168903615, 165840460734136218620282118980407, 434175962911910966534267888922363, 1136687428001596689926915871578146, 2975886321092879117718813750488296, 7790971535277040686646253728354427, 20397028284738242980109009807718891, 53400113318937688314986566416413837, 139803311672074822064045542536278117, 366009821697286778037650705008787602, 958226153419785512308602069401207274, 2508668638562069759308351643922323893, 6567779762266423766296344500004376663, 17194670648237201540680769634456908027, 45016232182445180857525943819371061607, 117854025899098341034777129018027092914, 308545845514849842251465489845085747444, 807783510645451185727159454321976495847, 2114804686421503714942213033535965616835, 5536630548619059959119219920505788577825, 14495086959435676162447387162616390216545, 37948630329687968528274622276198240394882, 99350804029628229422460100809468179391078, 260103781759196719739240982004551004524401
Generating function in Maple syntax:
-(3*x^3+x^2-3*x+1)/(x^2-3*x+1)/(x^2+x-1)
Generating function in latex syntax:
-\frac{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:
(-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)+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)+3 x^{3}+x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
1/40*(5+5^(1/2))*((-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-3*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)-2*(1/2*5^(1/2)-1/2)^(-n)-3*(-1/2*5^(1/2)-1/2)^(-n)+6*(3/2+1/2*5^(1/2))^(-n)+9*(3/2-1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\frac{\left(5+\sqrt{5}\right) \left(\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}-3 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}-2 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-3 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+6 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+9 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}\right)}{40}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = a(n)-2*a(n+1)-3*a(n+2)+4*a(n+3), n >= 4
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(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 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/23224/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[18,x]+F[8,x]
F[8,x] = F[14,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[9,x]
F[13,x] = x
F[14,x] = F[15,x]*F[17,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[13,x]*F[15,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[13,x]*F[23,x]
F[23,x] = F[24,x]+F[34,x]
F[24,x] = F[25,x]+F[27,x]
F[25,x] = F[26,x]
F[26,x] = F[13,x]*F[24,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[13,x]*F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[13,x]*F[32,x]
F[32,x] = F[27,x]+F[33,x]
F[33,x] = F[1,x]+F[30,x]
F[34,x] = F[35,x]+F[40,x]
F[35,x] = F[36,x]+F[37,x]+F[39,x]
F[36,x] = 0
F[37,x] = F[13,x]*F[38,x]
F[38,x] = F[25,x]+F[35,x]
F[39,x] = F[13,x]*F[34,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[42,x]
F[42,x] = F[13,x]*F[43,x]
F[43,x] = F[13,x]+F[41,x]
F[44,x] = F[45,x]
F[45,x] = F[13,x]*F[46,x]
F[46,x] = F[28,x]+F[44,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_{13}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{18}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{14}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \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_{9}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \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_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{34}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right) F_{34}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \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_{13}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{28}\! \left(x \right)+F_{44}\! \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_13(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_13(x)*F_7(x))
Eq(F_7(x), F_18(x) + F_8(x))
Eq(F_8(x), F_14(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(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_9(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x)*F_17(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_13(x)*F_15(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_13(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_34(x))
Eq(F_24(x), F_25(x) + F_27(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_13(x)*F_24(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_13(x)*F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_13(x)*F_32(x))
Eq(F_32(x), F_27(x) + F_33(x))
Eq(F_33(x), F_1(x) + F_30(x))
Eq(F_34(x), F_35(x) + F_40(x))
Eq(F_35(x), F_36(x) + F_37(x) + F_39(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_13(x)*F_38(x))
Eq(F_38(x), F_25(x) + F_35(x))
Eq(F_39(x), F_13(x)*F_34(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_13(x)*F_43(x))
Eq(F_43(x), F_13(x) + F_41(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_13(x)*F_46(x))
Eq(F_46(x), F_28(x) + F_44(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:
{"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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, 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