0123_0132_0213_0231_0312_1230
Counting sequence:
1, 1, 2, 6, 18, 55, 172, 551, 1806, 6043, 20588, 71232, 249700, 885062, 3166776, 11422417, 41487254, 151598803, 556904396, 2055439372, 7618111484, 28341512338, 105797174696, 396155102782, 1487577689228, 5600353098910, 21134070614392, 79928932401936, 302906067794376, 1150091999655404, 4374434019850736, 16665773291395141, 63591325883476966, 242995288464327555, 929796423130947852, 3562326387281978548, 13664824212759538828, 52477141473327214298, 201746335470278098760, 776401121421140337474, 2990812404445827884388, 11531714687706545485546, 44502149146318230771112, 171882836265550823418464, 664403263625832080562808, 2570177621348004199863956, 9949748365667682336354832, 38544730919537312852300234, 149420141278395070799779708, 579605505009259595584613678, 2249696075657614934161776568, 8737206751496842084361428952, 33952270765493838002027417560, 132008864765739174739554662132, 513530871593639578375753559440, 1998714795195973519099321821884, 7783017003047136265210951865560, 30321487623788196310560287591132, 118181986750993866343678110633328, 460833000908094962255248247922272, 1797715436446278299769480706961296, 7015801897412749150383140146428248, 27390936585297533868190349923736032, 106980376201860649523380248548521901, 417987958905039625775610228464103814, 1633732001768137581253627650691359203, 6387787611670565219235200261457679756, 24984355649327942100713611619517884612, 97752866838263447735889674380472122924, 382587113210778817039550413502657603498, 1497844253820368239181661057949128610696, 5865899717956493472359161748967922473242, 22978972084750457959299383855390584084084, 90043222679805599593367948400616220717618, 352933235330269164504784711623597678719176, 1383731857833787876404465799849888489059392, 5426578215481225361617332655597526311588504, 21286876951476106287618014895500831025169028, 83523120991228205795831845295868765364198096, 327798978661466278701778594191021184707367262, 1286802616531271488886312525182375620792899604, 5052626208331455815826884439891040279014618298, 19843616910250383773394059679690137284093775336, 77950794972035729067669131655723743621879942952, 306276743733522963582939763049194415014856988424, 1203646729443309721789519309598196803054460026588, 4731223577637991624176303594569516713512135838896, 18600955545750634412365397151603581112257462848612, 73144622081489854335603837695998478858890031482664, 287682196130120103251236986159309128022331954864868, 1131683937038356309754462978783623007696711790888336, 4452635064364008537591934470036353443529080453925472, 17522138084852251893405888008860892963886489765589744, 68965749293645145409435539243758097865488877862970088, 271490448857178353891990054913111964185333056527129376, 1068928881190971321050216990615039728087304945220141602, 4209346716351117617128008558322923645215935898837399004, 16578710177997330107997759750999326329563044112052068558, 65306362917742821509713334958106690129430416644439932216, 257292769846641852553887686942473554308574007591807444872, 1013831183264557860253494063775712889459922903144035315384
Generating function in Maple syntax:
((x-1)^2*(-4*x+1)^(1/2)+2*x^3-x^2+2*x-1)/(4*x^2-2*x)
Generating function in latex syntax:
\frac{\left(x -1\right)^{2} \sqrt{-4 x +1}+2 x^{3}-x^{2}+2 x -1}{4 x^{2}-2 x}
Generating function in sympy syntax:
(2*x**3 - x**2 + 2*x + sqrt(1 - 4*x)*(x - 1)**2 - 1)/(4*x**2 - 2*x)
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*F(x)^2-(x^2+1)*(2*x-1)^2*F(x)+x^5-2*x^3+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} F \! \left(x \right)^{2}-\left(x^{2}+1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{5}-2 x^{3}+5 x^{2}-4 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(n+3) = 4*(-1+2*n)/(n+4)*a(n)-2*(10+7*n)/(n+4)*a(1+n)+(19+7*n)/(n+4)*a(n+2), 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) = 18
a \! \left(n +3\right) = \frac{4 \left(-1+2 n \right) a \! \left(n \right)}{n +4}-\frac{2 \left(10+7 n \right) a \! \left(1+n \right)}{n +4}+\frac{\left(19+7 n \right) a \! \left(n +2\right)}{n +4}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22043/
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[19,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[11,x]
F[10,x] = F[1,x]+F[8,x]
F[11,x] = F[12,x]+F[6,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[8,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[8,x]
F[17,x] = F[12,x]
F[18,x] = F[6,x]*F[8,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[5,x]*F[8,x]
F[21,x] = F[22,x,1]
F[22,x,k[0]] = F[23,x,k[0]]+F[34,x,k[0]]
F[23,x,k[0]] = F[24,x]+F[27,x,k[0]]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[8,x]
F[27,x,k[0]] = F[28,x,k[0]]+F[32,x,k[0]]
F[28,x,k[0]] = F[29,x,k[0]]
F[29,x,k[0]] = F[30,x,k[0]]*F[31,x,k[0]]
F[30,x,k[0]] = k[0]*x
F[31,x,k[0]] = F[1,x]+F[28,x,k[0]]
F[32,x,k[0]] = F[33,x,k[0]]
F[33,x,k[0]] = F[27,x,k[0]]*F[8,x]
F[34,x,k[0]] = (F[35,x,k[0]]*k[0]-F[35,x,1])/(-1+k[0])
F[35,x,k[0]] = F[36,x,k[0]]
F[36,x,k[0]] = F[22,x,k[0]]*F[31,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{6}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x , 1\right)
F_{22}\! \left(x , y\right) = F_{23}\! \left(x , y\right)+F_{34}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{24}\! \left(x \right)+F_{27}\! \left(x , y\right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x , y\right) = F_{28}\! \left(x , y\right)+F_{32}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = F_{29}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{30}\! \left(x , y\right) F_{31}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = y x
F_{31}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{33}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{27}\! \left(x , y\right) F_{8}\! \left(x \right)
F_{34}\! \left(x , y\right) = \frac{y F_{35}\! \left(x , y\right)-F_{35}\! \left(x , 1\right)}{-1+y}
F_{35}\! \left(x , y\right) = F_{36}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = F_{22}\! \left(x , y\right) F_{31}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_8(x))
Eq(F_11(x), F_12(x) + F_6(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_8(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_8(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_6(x)*F_8(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_5(x)*F_8(x))
Eq(F_21(x), F_22(x, 1))
Eq(F_22(x, k_0), F_23(x, k_0) + F_34(x, k_0))
Eq(F_23(x, k_0), F_24(x) + F_27(x, k_0))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_8(x))
Eq(F_27(x, k_0), F_28(x, k_0) + F_32(x, k_0))
Eq(F_28(x, k_0), F_29(x, k_0))
Eq(F_29(x, k_0), F_30(x, k_0)*F_31(x, k_0))
Eq(F_30(x, k_0), k_0*x)
Eq(F_31(x, k_0), F_1(x) + F_28(x, k_0))
Eq(F_32(x, k_0), F_33(x, k_0))
Eq(F_33(x, k_0), F_27(x, k_0)*F_8(x))
Eq(F_34(x, k_0), (-k_0*F_35(x, k_0) + F_35(x, 1))/(1 - k_0))
Eq(F_35(x, k_0), F_36(x, k_0))
Eq(F_36(x, k_0), F_22(x, k_0)*F_31(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 3, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 3, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 3, 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"}}, {"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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": 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