0213_0231_1032_1230_1320_2031_2130
Counting sequence:
1, 1, 2, 6, 17, 45, 114, 280, 675, 1609, 3812, 9006, 21263, 50235, 118854, 281722, 669105, 1592329, 3796586, 9067956, 21692243, 51963573, 124628124, 299215402, 719014671, 1729098919, 4160828126, 10017836358, 24130420433, 58146123573, 140156924658, 337927458592, 814945149171, 1965681679057, 4742031994916, 11441185594998, 27607271631359, 66621447295683, 160781573237958, 388047359486434, 936601745464161, 2260701630429457, 5456906361684170, 13172316733359708, 31797144051698819, 76757812416358941, 185295182640802428, 447313002940928914, 1079850835335966687, 2606873961297045007, 6293317323683368190, 15192945724612920270, 36678082979634948449, 88546860056604274461, 213767299772383851762, 516072452853820104424, 1245894191812385170947, 3007824808864144942105, 7261471753860081179300, 17530624204492248956670, 42322431937477888944815, 102174911626801202498571, 246671102282690627734662, 595514810370393663989386, 1437696111371794888680657, 3470897809797502366645465, 8379473284312617052845674, 20229807485080035973729860, 48839014467731732223020915, 117907688846971695724777029, 284654097014386067362089372, 687215292580930379767062586, 1659083501586239140551009903, 4005379934572777165136332855, 9669838648369533571334418014, 23345047786585711477760554902, 56359915332086080927676442545, 136064840671843899164599313349, 328489521117939098010601002546, 793043731792041476813851415536, 1914576682470642926105756444883, 4622196492270540133292220175393, 11158968458086101967767211600484, 26940130990591425840855926698182, 65039225603566194581956188417311, 157018572526318098480016360862739, 379076351313390637491197392068486, 915171236467475345969834286605938, 2209418746877092434053430793513473, 5334008575479161063538550252874593, 12877435588350414059878683325542218, 31088879133209984620833388947089772, 75055192616830368416486016564351011, 181199261890990682363593312013989677, 437453711447051639883020810976033276, 1056106674881573751208010444489708578, 2549667041403158680975244281266308799, 6155440718073810126630034899663843103, 14860548398322616857816968156160382718, 35876537356262719522184992383420538014, 86613622793935406991425290576459825601
Generating function in Maple syntax:
(2*x^6-2*x^5-8*x^4+2*x^3+7*x^2-5*x+1)/(x-1)/(2*x-1)/(x^2+x-1)/(x^2+2*x-1)
Generating function in latex syntax:
\frac{2 x^{6}-2 x^{5}-8 x^{4}+2 x^{3}+7 x^{2}-5 x +1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(2*x**6 - 2*x**5 - 8*x**4 + 2*x**3 + 7*x**2 - 5*x + 1)/((x - 1)*(2*x - 1)*(x**2 + x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)*(x^2+x-1)*(x^2+2*x-1)*F(x)-2*x^6+2*x^5+8*x^4-2*x^3-7*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right) \left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)-2 x^{6}+2 x^{5}+8 x^{4}-2 x^{3}-7 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/20*(12*5^(1/2)-20)*(-1/2*5^(1/2)-1/2)^(-n)+1/20*(-12*5^(1/2)-20)*(1/2*5^(1/2)-1/2)^(-n)+1/20*(-10*2^(1/2)-5)*(-1-2^(1/2))^(-n)+1/20*(10*2^(1/2)-5)*(2^(1/2)-1)^(-n)+2^n+3/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(12 \sqrt{5}-20\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\frac{\left(-12 \sqrt{5}-20\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\\\frac{\left(-10 \sqrt{2}-5\right) \left(-1-\sqrt{2}\right)^{-n}}{20}+\frac{\left(10 \sqrt{2}-5\right) \left(\sqrt{2}-1\right)^{-n}}{20}+2^{n}+\frac{3}{2} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 45
a(6) = 114
a(n+5) = 2*a(n)+5*a(n+1)-3*a(n+2)-6*a(n+3)+5*a(n+4)-3, n >= 7
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) = 17
a \! \left(5\right) = 45
a \! \left(6\right) = 114
a \! \left(n +5\right) = 2 a \! \left(n \right)+5 a \! \left(n +1\right)-3 a \! \left(n +2\right)-6 a \! \left(n +3\right)+5 a \! \left(n +4\right)-3, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/23161/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[25,x]+F[8,x]
F[8,x] = F[18,x]+F[9,x]
F[9,x] = F[10,x]*F[15,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]*F[14,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[14,x]*F[15,x]
F[18,x] = F[19,x]*F[22,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[14,x]*F[19,x]
F[22,x] = F[23,x]
F[23,x] = F[14,x]*F[24,x]
F[24,x] = 2*F[23,x]+F[1,x]
F[25,x] = F[2,x]+F[26,x]
F[26,x] = F[16,x]*F[20,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_{14}\! \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_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{25}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{18}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \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_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{1}\! \left(x \right)
F_{25}\! \left(x \right) = F_{2}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \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_14(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_14(x)*F_7(x))
Eq(F_7(x), F_25(x) + F_8(x))
Eq(F_8(x), F_18(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_15(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_13(x)*F_14(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_14(x)*F_15(x))
Eq(F_18(x), F_19(x)*F_22(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_14(x)*F_19(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_14(x)*F_24(x))
Eq(F_24(x), F_1(x) + 2*F_23(x))
Eq(F_25(x), F_2(x) + F_26(x))
Eq(F_26(x), F_16(x)*F_20(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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 0, 3, 1], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 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