0213_0321_1302_2301_3120
Counting sequence:
1, 1, 2, 6, 19, 51, 123, 277, 597, 1251, 2575, 5241, 10593, 21319, 42795, 85773, 171757, 343755, 687783, 1375873, 2752089, 5504559, 11009539, 22019541, 44039589, 88079731, 176160063, 352320777, 704642257, 1409285271, 2818571355, 5637143581, 11274288093, 22548577179, 45097155415, 90194311953, 180388625097, 360777251455, 721554504243, 1443109009893, 2886218021269, 5772436044099, 11544872089839, 23089744181401, 46179488364609, 92358976731111, 184717953464203, 369435906930477, 738871813863117, 1477743627728491, 2955487255459335, 5910974510921121, 11821949021844793, 23643898043692239, 47287796087387235, 94575592174777333, 189151184349557637, 378302368699118355, 756604737398239903, 1513209474796483113, 3026418949592969649, 6052837899185942839, 12105675798371889339, 24211351596743782461, 48422703193487568829, 96845406386975141691, 193690812773950287543, 387381625547900579377, 774763251095801163177, 1549526502191602330911, 3099053004383204666515, 6198106008766409337861, 12396212017532818680693, 24792424035065637366499, 49584848070131274738255, 99169696140262549481913, 198339392280525098969377, 396678784561050197944455, 793357569122100395894763, 1586715138244200791795533, 3173430276488401583597229, 6346860552976803167200779, 12693721105953606334408039, 25387442211907212668822721, 50774884423814425337652249, 101549768847628850675311471, 203099537695257701350630083, 406199075390515402701267477, 812398150781030805402542437, 1624796301562061610805092531, 3249592603124123221610192895, 6499185206248246443220393801, 12998370412496492886440795793, 25996740824992985772881599959, 51993481649985971545763208475, 103986963299971943091526425693, 207973926599943886183052860317, 415947853199887772366105729755, 831895706399775544732211468823, 1663791412799551089464422947153, 3327582825599102178928845904009
Generating function in Maple syntax:
(x^6-2*x^5+2*x^4-2*x^3+6*x^2-4*x+1)/(x-1)^3/(2*x-1)
Generating function in latex syntax:
\frac{x^{6}-2 x^{5}+2 x^{4}-2 x^{3}+6 x^{2}-4 x +1}{\left(x -1\right)^{3} \left(2 x -1\right)}
Generating function in sympy syntax:
(x**6 - 2*x**5 + 2*x**4 - 2*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)^3*(2*x-1)*F(x)+x^6-2*x^5+2*x^4-2*x^3+6*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{3} \left(2 x -1\right) F \! \left(x \right)+x^{6}-2 x^{5}+2 x^{4}-2 x^{3}+6 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,-3+21/8*2^n-n-n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ -3+\frac{21 \,2^{n}}{8}-n -n^{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) = 19
a(5) = 51
a(6) = 123
a(n+1) = n^2+2*a(n)-n+1, 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) = 19
a \! \left(5\right) = 51
a \! \left(6\right) = 123
a \! \left(n +1\right) = n^{2}+2 a \! \left(n \right)-n +1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21937/
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[16,x]*F[4,x]
F[4,x] = F[22,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[8,x] = F[16,x]*F[9,x]
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]*F[17,x]
F[12,x] = F[13,x]+F[19,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]*F[17,x]
F[16,x] = x
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[16,x]*F[17,x]
F[19,x] = F[20,x]
F[20,x] = F[16,x]*F[17,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]
F[23,x] = F[16,x]*F[24,x]
F[24,x] = F[25,x]+F[40,x]
F[25,x] = F[26,x]+F[38,x]
F[26,x] = F[27,x]+F[34,x]
F[27,x] = F[13,x]*F[28,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]^2*F[16,x]
F[31,x] = F[1,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[16,x]*F[31,x]
F[34,x] = F[35,x]
F[35,x] = F[16,x]*F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[16,x]^2*F[17,x]^2*F[13,x]*F[31,x]
F[38,x] = F[17,x]*F[31,x]*F[39,x]
F[39,x] = F[19,x]
F[40,x] = F[41,x]*F[44,x]
F[41,x] = F[42,x]+F[7,x]
F[42,x] = F[39,x]*F[43,x]
F[43,x] = F[18,x]
F[44,x] = F[31,x]+F[32,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \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_{8}\! \left(x \right) = F_{16}\! \left(x \right) F_{9}\! \left(x \right)
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_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{40}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{38}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31} \left(x \right)^{2} F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16} \left(x \right)^{2} F_{17} \left(x \right)^{2} F_{13}\! \left(x \right) F_{31}\! \left(x \right)
F_{38}\! \left(x \right) = F_{17}\! \left(x \right) F_{31}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \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_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{39}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)
F_{44}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_22(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))
Eq(F_8(x), F_16(x)*F_9(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_16(x)*F_17(x))
Eq(F_12(x), F_13(x) + F_19(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x)*F_17(x))
Eq(F_16(x), x)
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_16(x)*F_17(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_16(x)*F_17(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_16(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_40(x))
Eq(F_25(x), F_26(x) + F_38(x))
Eq(F_26(x), F_27(x) + F_34(x))
Eq(F_27(x), F_13(x)*F_28(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_16(x)*F_31(x)**2)
Eq(F_31(x), F_1(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_16(x)*F_31(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_16(x)*F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_13(x)*F_16(x)**2*F_17(x)**2*F_31(x))
Eq(F_38(x), F_17(x)*F_31(x)*F_39(x))
Eq(F_39(x), F_19(x))
Eq(F_40(x), F_41(x)*F_44(x))
Eq(F_41(x), F_42(x) + F_7(x))
Eq(F_42(x), F_39(x)*F_43(x))
Eq(F_43(x), F_18(x))
Eq(F_44(x), F_31(x) + F_32(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, 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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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, 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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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, 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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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"}}, {"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, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 1], [0, 0], [0, 0], [0, 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", 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