0213_1302_1320_2031
Counting sequence:
1, 1, 2, 6, 20, 68, 232, 793, 2719, 9366, 32451, 113181, 397532, 1406291, 5009981, 17969617, 64866850, 235560383, 860163419, 3156915664, 11640177613, 43101643355, 160214782052, 597637189171, 2236481705485, 8393946034088, 31588816666027, 119171348735693, 450606079459404, 1707393202793167, 6482075597709665, 24653496625378129, 93923207077019102, 358384125263825891, 1369500950039703115, 5240516135463123188, 20079274852395517417, 77028515974624264443, 295838527209240492584, 1137444711809059610899, 4377771907034478848153, 16865535516379290849940, 65035427370732041392867, 251005365977093297070881, 969573831743373839586416, 3748233744144282908766007, 14501220404068152130250485, 56143543802184873588322298, 217519761730273740339849329, 843312190083914754529850461, 3271573975918396900764259366, 12699672092673271025746194881, 49327037906845409356155110477, 191700979192100139392988147406, 745419874785843611769311149405, 2900053825718342624012206157933, 11288369296996420015832006036762, 43961093066356022332629715936393, 171281464477644247207065018759169, 667651374915909096952212146084066, 2603633634611049114875057311797701, 10157675558309853812786495078447581, 39644874866194427848221724775385682, 154793324112954155868391147101542966, 604622566230940739660089989350029916, 2362534607979154413026350629765318232, 9234825695964995896667897817307339180, 36110304332751725270348064024515359058, 141247561257337430713844829691825946594, 552680033951555970975598937178618888424, 2163243643997180801589326572124143013878, 8469776605845527452231436618808416780360, 33171836713956590299196101464565616920202, 129955442121682821560707108270728420813346, 509264389232040439838045039601730241778076, 1996239623382610667507116149504806167257422, 7827072102418337850354168173291628459436670, 30697301549955679636109977232667895482253588, 120423790705361390367243192821184296840159614, 472533664518328474000523143551081103823946284, 1854636618446430504657391422777744936770508718, 7280947084914610978571626668278951976652006730, 28590209551937538300866146504962500262582436272, 112290952672444644962444583845445884681288039626, 441131217239231125199333964251206425915982550846, 1733341499888732330551309040373312718948420456552, 6812265950138942929023215668952686271911823324090, 26778611038080473784558091270905817323297137282018, 105286198080534539201009405708385174873925942993084, 414037620335998111601430291660798991459664993201834, 1628512947119812984484773838252870369694435382510018, 6406544253006735963411946294788077898672750654450420, 25207853167224274942043684864363007538789684348629242, 99203176139022725799925017951573907145638169043204106, 390473267697240076123722018452176139120466192454336676, 1537203733893971700086015354612940661040429511578250772, 6052633542435772993992982845856013310035867523189301840, 23835739840731013947782069946853638711361114614132021248, 93882085337099475757507171909433693634642298042163862704, 369832630825594047422183584961050583480565998864387392204, 1457118092641355463872574572256440981315809854886662128548
Generating function in Maple syntax:
1/2*((2*x-1)*(-4*x+1)^(1/2)-2*x^2-2*x+1)/(x^2-3*x+1)/x
Generating function in latex syntax:
\frac{\left(2 x -1\right) \sqrt{-4 x +1}-2 x^{2}-2 x +1}{2 \left(x^{2}-3 x +1\right) x}
Generating function in sympy syntax:
(-x**2 - x + sqrt(1 - 4*x)*(2*x - 1)/2 + 1/2)/(x*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^2*x*F(x)^2+(x^2-3*x+1)*(2*x^2+2*x-1)*F(x)+x^3+6*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right)^{2} x F \! \left(x \right)^{2}+\left(x^{2}-3 x +1\right) \left(2 x^{2}+2 x -1\right) F \! \left(x \right)+x^{3}+6 x^{2}-5 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -4*(3+2*n)/(5+n)*a(n)+2*(28+15*n)/(5+n)*a(1+n)-(77+27*n)/(5+n)*a(n+2)+(35+9*n)/(5+n)*a(n+3), n >= 4
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(n +4\right) = -\frac{4 \left(3+2 n \right) a \! \left(n \right)}{5+n}+\frac{2 \left(28+15 n \right) a \! \left(1+n \right)}{5+n}-\frac{\left(77+27 n \right) a \! \left(n +2\right)}{5+n}+\frac{\left(35+9 n \right) a \! \left(n +3\right)}{5+n}, \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21392/
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[5,x]+F[9,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]^2*F[8,x]
F[8,x] = x
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[8,x]
F[11,x] = F[12,x]+F[13,x]
F[12,x] = F[4,x]*F[5,x]
F[13,x] = F[14,x]*F[17,x]*F[6,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[8,x]
F[17,x] = F[18,x]+F[25,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[8,x]
F[21,x] = F[14,x]+F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[22,x]*F[8,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[30,x]
F[27,x] = 0
F[28,x] = F[29,x]*F[8,x]
F[29,x] = F[19,x]+F[26,x]
F[30,x] = F[31,x]*F[8,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[15,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[32,x]*F[8,x]
F[35,x] = F[26,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,x]*F[8,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_{5}\! \left(x \right)+F_{9}\! \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_{5} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \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_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{14}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right) F_{8}\! \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_5(x) + F_9(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_5(x)**2*F_8(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_8(x))
Eq(F_11(x), F_12(x) + F_13(x))
Eq(F_12(x), F_4(x)*F_5(x))
Eq(F_13(x), F_14(x)*F_17(x)*F_6(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_8(x))
Eq(F_17(x), F_18(x) + F_25(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_8(x))
Eq(F_21(x), F_14(x) + F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_22(x)*F_8(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_30(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_29(x)*F_8(x))
Eq(F_29(x), F_19(x) + F_26(x))
Eq(F_30(x), F_31(x)*F_8(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_15(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_32(x)*F_8(x))
Eq(F_35(x), F_26(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(x)*F_8(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, 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, 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, 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"}}, {"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, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 1], [0, 1]]}], "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": [[1, 2]]}, {"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, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[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": "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]]}], "requirements": []}, {"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, 1], 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