0213_0231_1320
Counting sequence:
1, 1, 2, 6, 21, 77, 285, 1054, 3889, 14330, 52800, 194748, 719602, 2664989, 9894443, 36831886, 137465657, 514370974, 1929432068, 7254441076, 27336758102, 103229845714, 390594944522, 1480679644876, 5622915464986, 21388666960212, 81486623660720, 310907347188280, 1187905972634884, 4544713448494301, 17408992511383847, 66765946853123566, 256344622615013385, 985273248301378102, 3790789093063333948, 14598954329955899076, 56274709436156909358, 217112919334927783558, 838342902969413573366, 3239702846099037632772, 12529132125475086423294, 48490307418234029239148, 187799184741084054266656, 727821309675593309836232, 2822515621216666429851004, 10952603676611513916489458, 42526277231316305632925326, 165213854931265430071704044, 642207458872005512885927434, 2497670769703614419930707820, 9718912168759472402566905160, 37836838139771977118703225608, 147373444242194680107231990556, 574280051870837218222614989444, 2238832335487169179360595909332, 8731851231255606385919911870424, 34069980807474150909546906324244, 132987993511888215128459136934952, 519304894011115717037234282006560, 2028600583201491236805341468911472, 7927381857655406459727702263490184, 30989694404552903569296279624530141, 121186568042503304031180941364811423, 474063975611790683714732350245427886, 1855070776241969134953326352097149097, 7261411387467166111880500729388710694, 28432481474839934168742755227767614188, 111362237717944991295239942975298184676, 436301838186992073789195458104856908702, 1709852459005604730370962368575595265854, 6702695039301259812711840785677046208958, 26281876350200331274888737028439534181716, 103080457680561765587609864990003657592022, 404395641208057760731319902611444721786364, 1586879520290965053541471547566669978437568, 6228537959462809014594777650716214221194536, 24452899067163190844306186817761136749715532, 96022758414591545889494419582502433808023014, 377150924158405111240666894463959395910133418, 1481668439912906248622230710373404183900311716, 5822097863203352597845039237021683903717534382, 22882234291272898229950193914159464953098271684, 89950943833032553214667508599938839785613393848, 353670939375521025748588072485282205152874267416, 1390840840347457576836388129064094870769427039572, 5470638515758933375809523025938507430444158736444, 21521835517794475833022342069358642980546394251148, 84683644720315967041494618032076577589029450811816, 333270609233758914371248657791841805092397461554732, 1311807411552940404852498974464326067525492453287064, 5164367492560581062559483849288374497661709493764128, 20334647272672394414635728789938768483603440246224656, 80080558798734807070811764145616013820880354066054712, 315418411154776675125709389456533484423275504546191474, 1242553418296280659146273363101281183960301890665203798, 4895643826921405464466339946485494990752467173774775596, 19291675588156785571283926246487957800450791005662420842, 76031610449542963461764559914810516681995712609387403324, 299696231794233660374018703556335082868221611993369591768, 1181489905028273744404063388335977810798492121516612367976, 4658426087404265209538135482373034333188400948196445439852
Generating function in Maple syntax:
((-5*x^3+8*x^2-5*x+1)*(1-4*x)^(1/2)-2*x^4+13*x^3-16*x^2+7*x-1)/(4*x^4-2*x^3)
Generating function in latex syntax:
\frac{\left(-5 x^{3}+8 x^{2}-5 x +1\right) \sqrt{1-4 x}-2 x^{4}+13 x^{3}-16 x^{2}+7 x -1}{4 x^{4}-2 x^{3}}
Generating function in sympy syntax:
(-2*x**4 + 13*x**3 - 16*x**2 + 7*x + sqrt(1 - 4*x)*(-5*x**3 + 8*x**2 - 5*x + 1) - 1)/(4*x**4 - 2*x**3)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*x^3*F(x)^2+(x^3-6*x^2+5*x-1)*(2*x-1)^2*F(x)+x^5+12*x^4-28*x^3+23*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} x^{3} F \! \left(x \right)^{2}+\left(x^{3}-6 x^{2}+5 x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{5}+12 x^{4}-28 x^{3}+23 x^{2}-8 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(n+5) = 20*(1+2*n)/(8+n)*a(n)-2*(117+23*n)/(8+n)*a(3+n)-2*(95+47*n)/(8+n)*a(n+1)+(334+93*n)/(8+n)*a(n+2)+(72+11*n)/(8+n)*a(n+4), n >= 6
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(5\right) = 77
a \! \left(n +5\right) = \frac{20 \left(1+2 n \right) a \! \left(n \right)}{8+n}-\frac{2 \left(117+23 n \right) a \! \left(3+n \right)}{8+n}-\frac{2 \left(95+47 n \right) a \! \left(n +1\right)}{8+n}+\frac{\left(334+93 n \right) a \! \left(n +2\right)}{8+n}+\frac{\left(72+11 n \right) a \! \left(n +4\right)}{8+n}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/310/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]*F[9,x]
F[7,x] = F[13,x]+F[8,x]
F[8,x] = F[0,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[9,x]^2*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]*F[9,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]*F[9,x]
F[16,x] = F[14,x]+F[17,x]
F[17,x] = F[18,x]*F[9,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = 2*F[20,x]+F[1,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_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{12}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{0}\! \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_{9} \left(x \right)^{2} F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{9}\! \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_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{1}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_12(x)*F_7(x)*F_9(x))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_0(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)**2)
Eq(F_12(x), x)
Eq(F_13(x), F_14(x)*F_9(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x)*F_9(x))
Eq(F_16(x), F_14(x) + F_17(x))
Eq(F_17(x), F_18(x)*F_9(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_1(x) + 2*F_20(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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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, 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": [1, 3, 2, 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, 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": [1, 3, 2, 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"}}, {"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, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 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": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "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": [[2, 1]]}, {"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, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"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": [1, 3, 2, 0], "pos": [[0, 0], [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, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 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": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 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], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 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": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, 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