0132_0231_0312_1023_1203
Counting sequence:
1, 1, 2, 6, 19, 59, 187, 610, 2037, 6927, 23910, 83578, 295309, 1053112, 3785718, 13704319, 49915782, 182803827, 672734021, 2486535096, 9226834594, 34360340187, 128371824008, 481025637316, 1807368057002, 6807866638037, 25702680415085, 97246672884209, 368666707643029, 1400220997329296, 5327326394474737, 20301296846638329, 77481178891649634, 296132991169667371, 1133337282256836634, 4342903152823316759, 16661697294801935541, 63995356892521944280, 246060572628696954608, 947057014821161652619, 3648620749396464072490, 14069524136467548836240, 54301174258335568976578, 209749301321701126357517, 810843197165112952695521, 3136916393604540861341095, 12144632605885352792566533, 47050881431419996181732504, 182406581534523386229298247, 707604826756960683977953264, 2746676446027134990261916161, 10667931047706082589563903685, 41457110186359825276451618226, 161196158412276784968414886324, 627102321191201180915565490227, 2440854489273270694703200044310, 9505112881216222354011325984324, 37031997224476465065715069966106, 144342560147922138974769900406341, 562862595223872653522201587206221, 2195809847576219364155202222303531, 8569695180672883427257798177798841, 33458662151662905828352258804875866, 130682922537820260201391112480254038, 510611921354547565121869209130114851, 1995813353747061034818362721153007495, 7803707876849522157879013428057633535, 30523170699585616720505277314176477059, 119426720482880526401928521467140205836, 467425534044592871893021079682772518801, 1830031009046762850369644772011007332630, 7166972633919785372487136317165139364745, 28076351845040127464607309403171620769427, 110019518479134082886990870335817812152586, 431240538740783860385333091785381405638964, 1690779500635371691506959854680026846659581, 6630844230507826734430161159193252672087832, 26011306178610672309340462005647996765793316, 102062010633938834411157352148137938235878556, 400564031791540702332367316225413255002827045, 1572472295874199316976437441030811487264658017, 6174399533548364509603358076377330482624533711, 24249605063200440493818674381849869276601821209, 95259974117775098100760193373130405746360135884, 374291369078953733399526751732554041916499258545, 1470958863789399559424863710811481820471139538506, 5782031828488942413924058502796093344929139631649, 22732521541741645487979515679227689364981342349879, 89392265704254627886804792266760805127215831601072, 351589229909447269318202240343808077261862487897422, 1383096919095989961255858581004334728357329652940543, 5441884334632979096836763572964829358092674240856826, 21415287503832629217702356119781766904724603695549692, 84289745964682802280915195557598356489584290833271254, 331818190733069978585852308187372466987199546856926487, 1306467695846231475934397416737035015511071846215791375, 5144802132955032461616855127236779631357807437561728707, 20263232073952900209053080484284520360951482980086937153, 79821037959083282185293368436222388567380815433080928050, 314480156660159878947211075335843783927566176878681342621, 1239181731016275948150204768157584393370039178442785868370
Generating function in Maple syntax:
1/2*(x^3+x^2+x-1+(x^3+x^2-x+1)*(1-4*x)^(1/2))/x/(x^6+2*x^5-x^4+x^3+3*x^2-3*x+1)*(x-1)
Generating function in latex syntax:
\frac{\left(x^{3}+x^{2}+x -1+\left(x^{3}+x^{2}-x +1\right) \sqrt{1-4 x}\right) \left(x -1\right)}{2 x \left(x^{6}+2 x^{5}-x^{4}+x^{3}+3 x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(x - 1)*(x**3/2 + x**2/2 + x/2 + sqrt(1 - 4*x)*(x**3 + x**2 - x + 1)/2 - 1/2)/(x*(x**6 + 2*x**5 - x**4 + x**3 + 3*x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x^6+2*x^5-x^4+x^3+3*x^2-3*x+1)*F(x)^2-(x-1)*(x^3+x^2+x-1)*F(x)+(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{6}+2 x^{5}-x^{4}+x^{3}+3 x^{2}-3 x +1\right) F \! \left(x \right)^{2}-\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 59
a(6) = 187
a(7) = 610
a(8) = 2037
a(9) = 6927
a(10) = 23910
a(n+11) = -2*(5+2*n)/(12+n)*a(n)-(16+7*n)/(12+n)*a(1+n)+14*(n+5)/(12+n)*a(n+2)+(n+6)/(12+n)*a(n+3)-(184+33*n)/(12+n)*a(n+4)+2*(145+22*n)/(12+n)*a(n+5)-(36+n)/(12+n)*a(n+6)-2*(165+23*n)/(12+n)*a(n+7)+(478+55*n)/(12+n)*a(n+8)-(306+31*n)/(12+n)*a(n+9)+(98+9*n)/(12+n)*a(n+10), n >= 11
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) = 187
a \! \left(7\right) = 610
a \! \left(8\right) = 2037
a \! \left(9\right) = 6927
a \! \left(10\right) = 23910
a \! \left(n +11\right) = -\frac{2 \left(5+2 n \right) a \! \left(n \right)}{12+n}-\frac{\left(16+7 n \right) a \! \left(1+n \right)}{12+n}+\frac{14 \left(n +5\right) a \! \left(n +2\right)}{12+n}+\frac{\left(n +6\right) a \! \left(n +3\right)}{12+n}-\frac{\left(184+33 n \right) a \! \left(n +4\right)}{12+n}+\frac{2 \left(145+22 n \right) a \! \left(n +5\right)}{12+n}-\frac{\left(36+n \right) a \! \left(n +6\right)}{12+n}-\frac{2 \left(165+23 n \right) a \! \left(n +7\right)}{12+n}+\frac{\left(478+55 n \right) a \! \left(n +8\right)}{12+n}-\frac{\left(306+31 n \right) a \! \left(n +9\right)}{12+n}+\frac{\left(98+9 n \right) a \! \left(n +10\right)}{12+n}, \quad n \geq 11
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/21646/
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[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[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[21,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[14,x]+F[17,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[18,x] = F[19,x]
F[19,x] = F[20,x]*F[8,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[6,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[5,x]*F[8,x]
F[24,x] = F[25,x]+F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[8,x]
F[27,x] = F[28,x,1]
F[28,x,k[0]] = (F[29,x,k[0]]*k[0]-F[29,x,1])/(-1+k[0])
F[29,x,k[0]] = F[30,x,k[0]]+F[34,x,k[0]]
F[30,x,k[0]] = F[0,x]+F[31,x,k[0]]
F[31,x,k[0]] = F[32,x,k[0]]
F[32,x,k[0]] = F[29,x,k[0]]*F[33,x,k[0]]
F[33,x,k[0]] = k[0]*x
F[34,x,k[0]] = F[35,x,k[0]]
F[35,x,k[0]] = F[28,x,k[0]]*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_{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_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \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_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{6}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x , 1\right)
F_{28}\! \left(x , y\right) = \frac{y F_{29}\! \left(x , y\right)-F_{29}\! \left(x , 1\right)}{-1+y}
F_{29}\! \left(x , y\right) = F_{30}\! \left(x , y\right)+F_{34}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{31}\! \left(x , y\right)
F_{31}\! \left(x , y\right) = F_{32}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{29}\! \left(x , y\right) F_{33}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = y x
F_{34}\! \left(x , y\right) = F_{35}\! \left(x , y\right)
F_{35}\! \left(x , y\right) = F_{28}\! \left(x , y\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_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)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_21(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_17(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))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_8(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_6(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_5(x)*F_8(x))
Eq(F_24(x), F_25(x) + F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_8(x))
Eq(F_27(x), F_28(x, 1))
Eq(F_28(x, k_0), (-k_0*F_29(x, k_0) + F_29(x, 1))/(1 - k_0))
Eq(F_29(x, k_0), F_30(x, k_0) + F_34(x, k_0))
Eq(F_30(x, k_0), F_0(x) + F_31(x, k_0))
Eq(F_31(x, k_0), F_32(x, k_0))
Eq(F_32(x, k_0), F_29(x, k_0)*F_33(x, k_0))
Eq(F_33(x, k_0), k_0*x)
Eq(F_34(x, k_0), F_35(x, k_0))
Eq(F_35(x, k_0), F_28(x, k_0)*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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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], 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