0123_0213_0231_2013
Counting sequence:
1, 1, 2, 6, 20, 69, 244, 881, 3234, 12022, 45120, 170596, 648788, 2479058, 9509628, 36598498, 141246128, 546433953, 2118424888, 8227983473, 32010173958, 124715628853, 486550020968, 1900433894943, 7431033132718, 29085434212043, 113945138510244, 446764109282497, 1753052792878806, 6883686776855335, 27047954010602384, 106344512026780145, 418355246670588450, 1646678051785692022, 6484715073908928464, 25549272833652453276, 100706969501727400804, 397120936308220737934, 1566600185282527179980, 6182389758603464299326, 24406686766523917426472, 96384588141112458784428, 380754779940096860956804, 1504580214845026808252840, 5947182192261834492084276, 23514036860521067770396612, 92994602172557006733544508, 367872892205622520362780924, 1455601752411273159266325100, 5760864119550073252380670826, 22804932616179301796860230852, 90294673385590627008746039506, 357588841588374178236633068216, 1416415933854132317143294160994, 5611505555426016180814909092532, 22235487422766620509295785533230, 88123078555154153225566098501144, 349305692131120580622001677701950, 1384815938553557713241548663370012, 5490935763505102814849296053994434, 21775405442283957488376346745632216, 86367329541244473463849967197956326, 342605220356335514604423093913986852, 1359244850077363800754907164837626706, 5393351697679095440119323554045430560, 21403031566244868512551533831443510289, 84946524348790990657775910618889146568, 337184850214107589425296463552281143697, 1338569826825945781947007805363609439126, 5314507681401913832022123750072625977805, 21102438123835441026584672465708694380984, 83800853415188988823540168471597911617463, 332819785993802336759577511461309494642542, 1321945217796808779692523302192315475105671, 5251219756107678202750173885320244404295300, 20861627231084922168465435400403515859506429, 82885069574416560983673604351789348284776318, 329339255736735400385024642326377462459490227, 1308726008244639738219618770185379839170955696, 5201049883408076185211973896302951720836177365, 20671377005582108610375682863600154293077088234, 82164267034623260048770523349586388860356363673, 326611049113185551268158257033324083951121483452, 1298411109339844081129054322210561487952330325851, 5162097697143270788721248572843882071958917824450, 20524475954120990012530747513266866728844034741749, 81611063436756484183180822749559339110822598053408, 324531134119100804305220166511286285934451843537987, 1290605057526723903498301888546580223066255066399926, 5132859077000796692178483978715379235287386608376365, 20415200220154905324436570903932052366403149891549804, 81203665699725156534482323728561380326010892767484643, 323016493924159623273749340900304785735224092924514282, 1284991460894466947444483561887741151433085090525899321, 5112127625025773811933958271174171015053523346776870360, 20338947701265738421946932370075745221472300908718747399, 80924510080422257008884988250814420076078154865936018838, 322000067873267563295670765388446557926595083646019963827, 1281314164116462065996283557012273301101046805020191908716, 5098924484804923778915187528734221432809524741819816098105, 20291976345204852974925068520240612262320086808696492654494
Generating function in Maple syntax:
1/2*(-x*(-4*x+1)^(1/2)-12*x^2+11*x-2)/(4*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{-x \sqrt{-4 x +1}-12 x^{2}+11 x -2}{2 \left(4 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-6*x**2 - x*sqrt(1 - 4*x)/2 + 11*x/2 - 1)/((x - 1)**2*(4*x - 1))
Implicit equation for the generating function in Maple syntax:
(4*x-1)*(x-1)^4*F(x)^2+(4*x-1)*(3*x-2)*(x-1)^2*F(x)+9*x^3-14*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(4 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(4 x -1\right) \left(3 x -2\right) \left(x -1\right)^{2} F \! \left(x \right)+9 x^{3}-14 x^{2}+7 x -1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+2) = -2*(2*n+3)/(n+1)*a(n)+(4+5*n)/(n+1)*a(n+1)+4/(n+1), n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(n +2\right) = -\frac{2 \left(2 n +3\right) a \! \left(n \right)}{n +1}+\frac{\left(4+5 n \right) a \! \left(n +1\right)}{n +1}+\frac{4}{n +1}, \quad n \geq 3
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/21145/
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[5,x]
F[4,x] = x
F[5,x] = F[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[4,x]*F[8,x]
F[8,x] = F[9,x,1]
F[9,x,k[0]] = F[10,x,k[0]]+F[18,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]+F[15,x,k[0]]
F[11,x,k[0]] = F[0,x]+F[12,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]
F[13,x,k[0]] = F[10,x,k[0]]*F[14,x,k[0]]
F[14,x,k[0]] = k[0]*x
F[15,x,k[0]] = F[16,x,k[0]]
F[16,x,k[0]] = F[17,x,k[0]]*F[4,x]
F[17,x,k[0]] = (F[9,x,k[0]]*k[0]-F[9,x,1])/(-1+k[0])
F[18,x,k[0]] = F[19,x,k[0]]
F[19,x,k[0]] = F[20,x,k[0]]*F[23,x]*F[4,x]
F[20,x,k[0]] = F[1,x]+F[21,x,k[0]]
F[21,x,k[0]] = F[22,x,k[0]]
F[22,x,k[0]] = F[14,x,k[0]]*F[20,x,k[0]]
F[23,x] = F[24,x,1]
F[24,x,k[0]] = F[11,x,k[0]]+F[25,x,k[0]]
F[25,x,k[0]] = F[26,x,k[0]]
F[26,x,k[0]] = F[27,x,k[0]]*F[4,x]
F[27,x,k[0]] = (F[24,x,k[0]]*k[0]-F[24,x,1])/(-1+k[0])
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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x , 1\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{18}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)+F_{15}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = y x
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{17}\! \left(x , y\right) F_{4}\! \left(x \right)
F_{17}\! \left(x , y\right) = -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{20}\! \left(x , y\right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{20}\! \left(x , y\right)
F_{23}\! \left(x \right) = F_{24}\! \left(x , 1\right)
F_{24}\! \left(x , y\right) = F_{11}\! \left(x , y\right)+F_{25}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{26}\! \left(x , y\right)
F_{26}\! \left(x , y\right) = F_{27}\! \left(x , y\right) F_{4}\! \left(x \right)
F_{27}\! \left(x , y\right) = -\frac{-y F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}
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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_4(x)*F_8(x))
Eq(F_8(x), F_9(x, 1))
Eq(F_9(x, k_0), F_10(x, k_0) + F_18(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0) + F_15(x, k_0))
Eq(F_11(x, k_0), F_0(x) + F_12(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0))
Eq(F_13(x, k_0), F_10(x, k_0)*F_14(x, k_0))
Eq(F_14(x, k_0), k_0*x)
Eq(F_15(x, k_0), F_16(x, k_0))
Eq(F_16(x, k_0), F_17(x, k_0)*F_4(x))
Eq(F_17(x, k_0), (-k_0*F_9(x, k_0) + F_9(x, 1))/(1 - k_0))
Eq(F_18(x, k_0), F_19(x, k_0))
Eq(F_19(x, k_0), F_20(x, k_0)*F_23(x)*F_4(x))
Eq(F_20(x, k_0), F_1(x) + F_21(x, k_0))
Eq(F_21(x, k_0), F_22(x, k_0))
Eq(F_22(x, k_0), F_14(x, k_0)*F_20(x, k_0))
Eq(F_23(x), F_24(x, 1))
Eq(F_24(x, k_0), F_11(x, k_0) + F_25(x, k_0))
Eq(F_25(x, k_0), F_26(x, k_0))
Eq(F_26(x, k_0), F_27(x, k_0)*F_4(x))
Eq(F_27(x, k_0), (-k_0*F_24(x, k_0) + F_24(x, 1))/(1 - k_0))
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, 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": [2, 0, 1, 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, 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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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"}}, {"children": [{"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]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "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]]}]]}, 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