0123_0213_1302
Counting sequence:
1, 1, 2, 6, 21, 75, 265, 925, 3201, 11017, 37793, 129393, 442497, 1512225, 5165953, 17643457, 60250113, 205729921, 702452225, 2398414593, 8188884993, 27958972929, 95458646017, 325917686785, 1112755552257, 3799191029761, 12971261403137, 44286680330241, 151204232069121, 516243634724865, 1762566208978945, 6017777834901505, 20545979458519041, 70148363238014977, 239501496182505473, 817709262548959233, 2791834066420760577, 9531917757764993025, 32544002932578189313, 111112176283502247937, 379360699406291566593, 1295218445333039677441, 4422152383069331390465, 15098172642710757834753, 51548385806903391813633, 175997197946590098096129, 600892020181349701779457, 2051573684849810796969985, 7004510699071728156409857, 23914895426657659775877121, 81650560308627920279044097, 278772450381477836541132801, 951788680909218455559864321, 3249609822875044049064034305, 11094861929683991084950093825, 37880228072990379841299677185, 129331188432602544394553262081, 441564297584447432294123175937, 1507594813472620669184405143553, 5147250658721659869743412150273, 17573813007941542255792914169857, 60000750714323137514060984090625, 204855376841410042005410411446273, 699420005936995045915024284450817, 2387969270065162405492285528604673, 8153037068386664141825111972904961, 27836209733416340979687913689186305, 95038764796892054081845504520486913, 324482639720735571261494338122678273, 1107853029289158250669262638287945729, 3782446837715162007728014466582839297, 12914081292282331824721437769108291585, 44091431493699003873725532501973139457, 150537563390231353026050875187087278081, 513967390573527406717935677179225440257, 1754794435513646925542007441212372418561, 5991242960907532898176891376230329221121, 20455382972602837760513016553975152902145, 69839045968596285283477215326397114875905, 238445417929179465688440691923552477118465, 814103579779525292337924064493244325560321, 2779523483259742238277046329029529641680897, 9489886773479918369036800096938944502956033, 32400500127400189000802033549311347903168513, 110622226962640919267552385642596760956174337, 377687907595763299073441178750222864717185025, 1289507176457771357768331350272614970354040833, 4402652890639558832945785856703848218777092097, 15031597209642692616285166352497831067990884353, 51321083057291652799326464949038964101590548481, 175221137809881225964890269596070866804742815745, 598242385124941598261217633496026884084514947073, 2042527264880003941115708964811608492866023718913, 6973624289270132567941638532293665583569964105729, 23809442627320522389537612079630016109097607233537, 81290521930741824422272123014089874790350097219585, 277543202468326252910023171417413749985404367405057, 947591766011821362795568246682103816445315661168641, 3235280659110632945362266257974844897979250681839617, 11045939104418889055858007766697686223363964948971521, 37713195099454290332707657007166083626172545520107521
Generating function in Maple syntax:
(x^2-3*x+1)^2/(x-1)/(2*x-1)/(2*x^2-4*x+1)
Generating function in latex syntax:
\frac{\left(x^{2}-3 x +1\right)^{2}}{\left(x -1\right) \left(2 x -1\right) \left(2 x^{2}-4 x +1\right)}
Generating function in sympy syntax:
(x**2 - 3*x + 1)**2/((x - 1)*(2*x - 1)*(2*x**2 - 4*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)*(2*x^2-4*x+1)*F(x)-(x^2-3*x+1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right) \left(2 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x^{2}-3 x +1\right)^{2} = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1-1/8*(1+1/2*2^(1/2))^(-n)*2^(1/2)+1/8*(1-1/2*2^(1/2))^(-n)*2^(1/2)-1/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1-\frac{\left(1+\frac{\sqrt{2}}{2}\right)^{-n} \sqrt{2}}{8}+\frac{\left(1-\frac{\sqrt{2}}{2}\right)^{-n} \sqrt{2}}{8}-\frac{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) = 21
a(n+3) = 4*a(n)-10*a(n+1)+6*a(n+2)+1, n >= 5
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) = 21
a \! \left(n +3\right) = 4 a \! \left(n \right)-10 a \! \left(n +1\right)+6 a \! \left(n +2\right)+1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14632/
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[12,x]*F[4,x]
F[4,x] = F[16,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,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[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[46,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[25,x]
F[21,x] = F[22,x]+F[6,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[10,x]
F[25,x] = F[17,x]+F[26,x]
F[26,x] = 2*F[18,x]+F[27,x]+F[31,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[22,x]
F[30,x] = F[26,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[18,x]+F[34,x]+F[44,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[10,x]+F[37,x]
F[37,x] = F[23,x]
F[38,x] = F[33,x]+F[39,x]
F[39,x] = 2*F[18,x]+F[31,x]+F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[37,x]
F[43,x] = F[39,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[2,x]+F[33,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[45,x]+F[48,x]
F[48,x] = F[17,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[48,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{16}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{46}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)
F_{25}\! \left(x \right) = F_{17}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{22}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{23}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{37}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_16(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_12(x)*F_8(x))
Eq(F_8(x), F_13(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_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_46(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_25(x))
Eq(F_21(x), F_22(x) + F_6(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_10(x))
Eq(F_25(x), F_17(x) + F_26(x))
Eq(F_26(x), 2*F_18(x) + F_27(x) + F_31(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_22(x))
Eq(F_30(x), F_26(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_18(x) + F_34(x) + F_44(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_10(x) + F_37(x))
Eq(F_37(x), F_23(x))
Eq(F_38(x), F_33(x) + F_39(x))
Eq(F_39(x), 2*F_18(x) + F_31(x) + F_40(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_37(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_2(x) + F_33(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_45(x) + F_48(x))
Eq(F_48(x), F_17(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_48(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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]]}], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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]]}], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], 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