0213_1230_1302_1320_2031
Counting sequence:
1, 1, 2, 6, 19, 59, 180, 539, 1587, 4604, 13187, 37359, 104848, 291887, 806943, 2217440, 6061575, 16494323, 44703956, 120734283, 325064083, 872803388, 2337798723, 6248236703, 16667438304, 44384348831, 118009822079, 313330107264, 830887276231, 2200855430147, 5823683539812, 15395828612507, 40667274996339, 107338977850172, 283119899506499, 746295794660367, 1966091094634480, 5176906226004815, 13624821551965215, 35842900873626848, 94254811464580167, 247768839834394451, 651097457221259828, 1710462748825839531, 4492215797065545235, 11794975524253265276, 30962220845803358595, 81259504632327262271, 213221451553767973056, 559383011041605264959, 1467289114875452716031, 3848185666759307546112, 10091032296162216675463, 26458141798633855715555, 69363619508737053826884, 181826401811329085975675, 476583185879451188404851, 1249052047574577669114620, 3273298353804388065106691, 8577422726527949972639535, 22474854702161490520293712, 58885490953496333426531183, 154274202397961944217144607, 404160634725543744206449376, 1058747323853303341278276999, 2773384101198170887272345779, 7264550134002217102523303060, 19027859380611829619551694859, 49837359335536035991906750867, 130528474388875272137536955324, 341854906739890764440344534467, 895294187521958447446403194079, 2344650382338469343328833178528, 6140160354652302397431968387807, 16079460199334071959718842685823, 42106982675710921949196090072960, 110262642213778119795716492950855, 288732015177026847445423517839043, 756056700140255767911950319744036, 1979735750129384755937907960090203, 5183869176898808842362566599662963, 13573606698867843551307729920997308, 35541139383183909305515035336920003, 93059923296396408315659624459558223, 243663042661816245400973718835966384, 637988140848200100086125485768619151, 1670443664360517306473484414228777759, 4373686357355219649683042837204002784, 11451444702441117664724531483811168711, 29982649844590967438807792567564616467, 78501338315371457301823717712366741492, 205533034164341640739913348910610635627, 538125935787561723871907185996642867219, 1408912785481442512571771928977835568892, 3688776616833800491218790932917251090179, 9657813469658984207590257324045558492287, 25285620797601951122056532420567790591360, 66201359338710920026329811436275581227135, 173324035055132564963409820830141644676095, 453784211915484048305546809563130082899968, 1188061110705575303964887542072287913770247
Generating function in Maple syntax:
(x^6-x^5-5*x^4-3*x^3+11*x^2-6*x+1)/(2*x-1)/(x^2-3*x+1)/(x^2+2*x-1)
Generating function in latex syntax:
\frac{x^{6}-x^{5}-5 x^{4}-3 x^{3}+11 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(x**6 - x**5 - 5*x**4 - 3*x**3 + 11*x**2 - 6*x + 1)/((2*x - 1)*(x**2 - 3*x + 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2-3*x+1)*(x^2+2*x-1)*F(x)+x^6-x^5-5*x^4-3*x^3+11*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+x^{6}-x^{5}-5 x^{4}-3 x^{3}+11 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(8*5^(1/2)+20)*(3/2-1/2*5^(1/2))^(-n)+1/20*(-8*5^(1/2)+20)*(3/2+1/2*5^(1/2))^(-n)+1/20*(20*2^(1/2)-20)*(-1-2^(1/2))^(-n)+1/20*(-20*2^(1/2)-20)*(2^(1/2)-1)^(-n)+3/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(8 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(-8 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(20 \sqrt{2}-20\right) \left(-1-\sqrt{2}\right)^{-n}}{20}+\frac{\left(-20 \sqrt{2}-20\right) \left(\sqrt{2}-1\right)^{-n}}{20}+\frac{3 \,2^{n}}{4} & \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) = 59
a(6) = 180
a(n+5) = -2*a(n)+3*a(n+1)+11*a(n+2)-16*a(n+3)+7*a(n+4), 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) = 59
a \! \left(6\right) = 180
a \! \left(n +5\right) = -2 a \! \left(n \right)+3 a \! \left(n +1\right)+11 a \! \left(n +2\right)-16 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21954/
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[15,x]*F[18,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[14,x]*F[9,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[12,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[14,x]*F[15,x]*F[9,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[14,x]*F[24,x]
F[24,x] = F[19,x]+F[21,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[12,x]*F[2,x]
F[27,x] = F[16,x]*F[19,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_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \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_{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_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \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) = F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{27}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \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_15(x)*F_18(x)*F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_14(x)*F_9(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_12(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_14(x)*F_15(x)*F_9(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_14(x)*F_24(x))
Eq(F_24(x), F_19(x) + F_21(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_12(x)*F_2(x))
Eq(F_27(x), F_16(x)*F_19(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, 2, 3, 0], "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, 0, 3, 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, 2, 3, 0], "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, 0, 3, 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, 2, 3, 0], "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, 0, 3, 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": [1, 0, 2], "pos": [[0, 0], [1, 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": [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": [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, 0, 2], "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]]}], "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": [1, 0, 2], "pos": [[0, 0], [2, 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": [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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": 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