021_0123_1023_2103
Counting sequence:
1, 1, 2, 5, 11, 23, 49, 106, 228, 489, 1050, 2256, 4846, 10408, 22355, 48017, 103136, 221525, 475813, 1021999, 2195151, 4714962, 10127262, 21752337, 46721824, 100353760, 215549744, 462979088, 994432337, 2135940257, 4587784018, 9854096869, 21165605163, 45461582919, 97646890113, 209735661114, 450491024260, 967609236601, 2078326946234, 4464036443696, 9588299572766, 20594699406392, 44235334995619, 95013033381169, 204078402778800, 438339804536757, 941509643475525, 2022267655327839, 4343626746815647, 9329671700946850, 20039192849905982, 43042162998615329, 92450220399374144, 198573739246510784, 426516343043874400, 916114041936270112, 1967720467270529697, 4226464894186944321, 9078019870664273826, 19498670126308692165, 41881174761824184139, 89956534885105842295, 193217554535062902737, 415011799067098771466, 891403443022330419236, 1914644595691590864905, 4112463280803350474842, 8833155915208862623888, 18972727072507154438478, 40751502183728230161096, 87530112243951401661939, 188005843683915016422609, 403817570355546049907584, 867359369967327484414741, 1863000354362334600652517, 4001536664652535619389583, 8594896743344532305109359, 18460970427011938144541042, 39652300578353538374149342, 85169138175721946968340785, 182934709759440961861180512, 392925286689238394172011424, 843963844383842264862713232, 1812749127521760015067916592, 3893602102978682938965354481, 8363064202406045233963900897, 17963017535885171126962526450, 38582748043675944533855682725, 82871847317852332021744636523, 178000360941089392216418528423, 382326803620470000793763484193, 821199372820501117248345177562, 1763853341002530511052290674372, 3788578890264002746342744513689, 8137484945089564885695671039994, 17478496066620100889433450741744, 37542044847063233407167537335422, 80636521925393000071730109858904, 173199107686139567775498635271491, 372014196383988801326126392324721, 799048933681660936948853772726608
Generating function in Maple syntax:
-1/(x^2+1)/(x^3+2*x^2+x-1)
Generating function in latex syntax:
-\frac{1}{\left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right)}
Generating function in sympy syntax:
-1/((x**2 + 1)*(x**3 + 2*x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+1)*(x^3+2*x^2+x-1)*F(x)+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right) F \! \left(x \right)+1 = 0
Explicit closed form in Maple syntax:
-949/5184*(2^(1/3)*((I-41/403*31^(1/2))*3^(1/2)-123/403*I*31^(1/2)+1)*(29+3*31^(1/2)*3^(1/2))^(2/3)-16/13-4/13*((I+5/62*31^(1/2))*3^(1/2)-15/62*I*31^(1/2)-1)*2^(2/3)*(29+3*31^(1/2)*3^(1/2))^(1/3))*((199/73*2^(1/3)*((I+157/6169*31^(1/2))*3^(1/2)-1893/6169*I*31^(1/2)-53/199)*(29+3*31^(1/2)*3^(1/2))^(2/3)-46/73*2^(2/3)*((I-94/713*31^(1/2))*3^(1/2)-156/713*I*31^(1/2)+31/23)*(29+3*31^(1/2)*3^(1/2))^(1/3)+164/73+132/2263*I*31^(1/2))*(29/24*((I+3/29*31^(1/2))*3^(1/2)-9/29*I*31^(1/2)-1)*2^(1/3)*(29+3*31^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(116+12*31^(1/2)*3^(1/2))^(1/3)-1/12*(116+12*31^(1/2)*3^(1/2))^(1/3)-2/3)^(-n)+(((I-1025/2263*31^(1/2))*3^(1/2)-711/2263*I*31^(1/2)+325/73)*2^(1/3)*(29+3*31^(1/2)*3^(1/2))^(2/3)+8/73*((I-125/124*31^(1/2))*3^(1/2)-63/124*I*31^(1/2)+25/2)*2^(2/3)*(29+3*31^(1/2)*3^(1/2))^(1/3)+164/73-132/2263*I*31^(1/2))*(1/12*(-3*31^(1/2)*3^(1/2)+29)*2^(1/3)*(29+3*31^(1/2)*3^(1/2))^(2/3)+1/6*(116+12*31^(1/2)*3^(1/2))^(1/3)-2/3)^(-n)+136/73*2^(1/3)*cos(1/2*n*Pi)*((I-7/68*31^(1/2))*3^(1/2)-21/68*I*31^(1/2)+1)*(29+3*31^(1/2)*3^(1/2))^(2/3)-19/73*((I+1/19*31^(1/2))*3^(1/2)-3/19*I*31^(1/2)-1)*cos(1/2*n*Pi)*2^(2/3)*(29+3*31^(1/2)*3^(1/2))^(1/3)+200/73*cos(1/2*n*Pi)+72/73*(-29/24*((I-3/29*31^(1/2))*3^(1/2)-9/29*I*31^(1/2)+1)*2^(1/3)*(29+3*31^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(116+12*31^(1/2)*3^(1/2))^(1/3)-1/12*(116+12*31^(1/2)*3^(1/2))^(1/3)-2/3)^(-n))
Explicit closed form in latex syntax:
-\frac{949 \left(2^{\frac{1}{3}} \left(\left(i-\frac{41 \sqrt{31}}{403}\right) \sqrt{3}-\frac{123 i \sqrt{31}}{403}+1\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{16}{13}-\frac{4 \left(\left(i+\frac{5 \sqrt{31}}{62}\right) \sqrt{3}-\frac{15 i \sqrt{31}}{62}-1\right) 2^{\frac{2}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{13}\right) \left(\left(\frac{199 \,2^{\frac{1}{3}} \left(\left(i+\frac{157 \sqrt{31}}{6169}\right) \sqrt{3}-\frac{1893 i \sqrt{31}}{6169}-\frac{53}{199}\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{73}-\frac{46 \,2^{\frac{2}{3}} \left(\left(i-\frac{94 \sqrt{31}}{713}\right) \sqrt{3}-\frac{156 i \sqrt{31}}{713}+\frac{31}{23}\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{73}+\frac{164}{73}+\frac{132 i \sqrt{31}}{2263}\right) \left(\frac{29 \left(\left(i+\frac{3 \sqrt{31}}{29}\right) \sqrt{3}-\frac{9 i \sqrt{31}}{29}-1\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{i \sqrt{3}\, \left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{2}{3}\right)^{-n}+\left(\left(\left(i-\frac{1025 \sqrt{31}}{2263}\right) \sqrt{3}-\frac{711 i \sqrt{31}}{2263}+\frac{325}{73}\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{8 \left(\left(i-\frac{125 \sqrt{31}}{124}\right) \sqrt{3}-\frac{63 i \sqrt{31}}{124}+\frac{25}{2}\right) 2^{\frac{2}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{73}+\frac{164}{73}-\frac{132 i \sqrt{31}}{2263}\right) \left(\frac{\left(-3 \sqrt{31}\, \sqrt{3}+29\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}+\frac{\left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{2}{3}\right)^{-n}+\frac{136 \,2^{\frac{1}{3}} \cos \left(\frac{n \pi}{2}\right) \left(\left(i-\frac{7 \sqrt{31}}{68}\right) \sqrt{3}-\frac{21 i \sqrt{31}}{68}+1\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{73}-\frac{19 \left(\left(i+\frac{\sqrt{31}}{19}\right) \sqrt{3}-\frac{3 i \sqrt{31}}{19}-1\right) \cos \left(\frac{n \pi}{2}\right) 2^{\frac{2}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{73}+\frac{200 \cos \left(\frac{n \pi}{2}\right)}{73}+\frac{72 \left(-\frac{29 \left(\left(i-\frac{3 \sqrt{31}}{29}\right) \sqrt{3}-\frac{9 i \sqrt{31}}{29}+1\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{i \sqrt{3}\, \left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{2}{3}\right)^{-n}}{73}\right)}{5184}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 5
a(4) = 11
a(n+5) = a(n)+2*a(n+1)+2*a(n+2)+a(n+3)+a(n+4), 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) = 5
a \! \left(4\right) = 11
a \! \left(n +5\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/5485/
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[10,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[1,x]+F[4,x]
F[10,x] = F[11,x]+F[2,x]
F[11,x] = F[12,x]+F[13,x]+F[25,x]
F[12,x] = 0
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[15,x]+F[19,x]
F[15,x] = F[16,x]+F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[4,x]
F[19,x] = F[20,x]+F[22,x]
F[20,x] = F[21,x]
F[21,x] = F[2,x]*F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[20,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[2,x]+F[27,x]
F[27,x] = F[12,x]+F[13,x]+F[21,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_{10}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{2}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{25}\! \left(x \right)
F_{12}\! \left(x \right) = 0
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \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_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \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_{20}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{21}\! \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_10(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_1(x) + F_4(x))
Eq(F_10(x), F_11(x) + F_2(x))
Eq(F_11(x), F_12(x) + F_13(x) + F_25(x))
Eq(F_12(x), 0)
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_19(x))
Eq(F_15(x), F_16(x) + F_4(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_4(x))
Eq(F_19(x), F_20(x) + F_22(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_2(x)*F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_20(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_2(x) + F_27(x))
Eq(F_27(x), F_12(x) + F_13(x) + F_21(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, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], 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