0213_1032_1320_2301_3021
Counting sequence:
1, 1, 2, 6, 19, 50, 121, 288, 696, 1723, 4357, 11183, 28967, 75421, 196914, 514840, 1347011, 3525468, 9228521, 24159058, 63247432, 165581813, 433496357, 1134905361, 2971217559, 7778744855, 20365014226, 53316294698, 139583866371, 365435300518, 956722030857, 2504730787268, 6557470325672, 17167680183951, 44945570219829, 117669030468595, 308061521178391, 806515533058353, 2111485077987746, 5527939700895228, 14472334024687507, 37889062373156048, 99194853094768537, 259695496911136566, 679891637638627224, 1779979416004730185, 4660046610375547381, 12200160415121894933, 31940434634990119271, 83621143489848443563, 218922995834555190882, 573147844013817107278, 1500520536206896107827, 3928413764606871191706, 10284720757613717441369, 26925748508234281105000, 70492524767089125844696, 184551825793033096398563, 483162952612010163318821, 1264937032042997393524023, 3311648143516982017217607, 8670007398507948658091333, 22698374052006863957017042, 59425114757512643212918496, 155576970220531065681695139, 407305795904080553832121540, 1066340417491710595814621961, 2791715456571051233611694618, 7308805952221443105020409896, 19134702400093278081449480733, 50095301248058391139327975557, 131151201344081895336534386713, 343358302784187294870275122807, 898923707008479989274290917311, 2353412818241252672952597562034, 6161314747715278029583501698930, 16130531424904581415797907462051, 42230279526998466217810220611598, 110560307156090817237632754294121, 289450641941273985495088042189068, 757791618667731139247631372188232, 1983924214061919432247806074287543, 5193981023518027157495786850582997, 13598018856492162040239554477366651, 35600075545958458963222876581418679, 93202207781383214849429075266787545, 244006547798191185585064349218838466, 638817435613190341905763972389618628, 1672445759041379840132227567949904371, 4378519841510949178490918731459977528, 11463113765491467695340528626429907257, 30010821454963453907530667147829619198, 78569350599398894027251472817058821112, 205697230343233228174223751303346710641, 538522340430300790495419781092981172949, 1409869790947669143312035591975596665885, 3691087032412706639440686994833808677831, 9663391306290450775010025392525829216083, 25299086886458645685589389182743678814146, 66233869353085486281758142155705207065238, 173402521172797813159685037284371942215507
Generating function in Maple syntax:
-(2*x^7-8*x^6+9*x^5-8*x^4+15*x^3-14*x^2+6*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
-\frac{2 x^{7}-8 x^{6}+9 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-2*x**7 + 8*x**6 - 9*x**5 + 8*x**4 - 15*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^4*F(x)+2*x^7-8*x^6+9*x^5-8*x^4+15*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+2 x^{7}-8 x^{6}+9 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/30*(-3*5^(1/2)+15)*(3/2-1/2*5^(1/2))^(-n)+1/30*(3*5^(1/2)+15)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3+1/2*n^2-14/3*n+6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-3 \sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(3 \sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}+\frac{n^{2}}{2}-\frac{14 n}{3}\\+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) = 50
a(6) = 121
a(7) = 288
a(n+2) = -a(n)+3*a(n+1)-1/6*n*(n^2+6*n-25), 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) = 50
a \! \left(6\right) = 121
a \! \left(7\right) = 288
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{n \left(n^{2}+6 n -25\right)}{6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21953/
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[8,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[8,x]*F[9,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[12,x]*F[8,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,x]
F[16,x] = F[17,x]+F[22,x]
F[17,x] = F[18,x]+F[20,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[5,x]+F[6,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]^2*F[6,x]*F[8,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[30,x]*F[8,x]
F[24,x] = F[25,x]+F[35,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]+F[32,x]
F[27,x] = F[28,x]*F[30,x]
F[28,x] = F[12,x]+F[29,x]
F[29,x] = F[13,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[30,x]*F[8,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]^2*F[34,x]*F[8,x]
F[34,x] = F[31,x]
F[35,x] = F[28,x]*F[34,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{22}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{20}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{5}\! \left(x \right)+F_{6}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12} \left(x \right)^{2} F_{6}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12} \left(x \right)^{2} F_{34}\! \left(x \right) F_{8}\! \left(x \right)
F_{34}\! \left(x \right) = F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right) F_{34}\! \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_8(x))
Eq(F_4(x), F_14(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_8(x)*F_9(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_12(x)*F_8(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(x))
Eq(F_16(x), F_17(x) + F_22(x))
Eq(F_17(x), F_18(x) + F_20(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_5(x) + F_6(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)**2*F_6(x)*F_8(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_30(x)*F_8(x))
Eq(F_24(x), F_25(x) + F_35(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x) + F_32(x))
Eq(F_27(x), F_28(x)*F_30(x))
Eq(F_28(x), F_12(x) + F_29(x))
Eq(F_29(x), F_13(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_30(x)*F_8(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)**2*F_34(x)*F_8(x))
Eq(F_34(x), F_31(x))
Eq(F_35(x), F_28(x)*F_34(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": [1, 0, 3, 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, 1, 3], "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, 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, 1, 3], "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, 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": []}, "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": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "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], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 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": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "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], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], 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