0213_0231_1032_1302_3021

Counting sequence:
1, 1, 2, 6, 19, 58, 177, 554, 1790, 5951, 20241, 70075, 245997, 873285, 3128951, 11298882, 41076185, 150206040, 552108207, 2038688441, 7558900482, 28130070240, 105035554295, 393391365807, 1477484652090, 5563288966964, 20997298103861, 79422040687139, 301020239352174, 1143051699420418, 4348068331271655, 16566753181876638, 63218472537419678, 241587975548373687, 924472887059571377, 3542147603463224671, 13588191998549368309, 52185605265100370809, 200635403812034648687, 772161277029981261867, 2974607792890148692363, 11469696981871470288859, 44264494269397719144435, 170971034778569499160901, 660901009993745920121937, 2556710890801355392875483, 9897913776376547600696699, 38345021945638933518375615, 148649991626755932071381541, 576632928732611673719226389, 2238213129646343939451013839, 8692813249605108021775116657, 33780513261746158419005376467, 131343855396258612902998710897, 510954301413566671853434903923, 1988725251440727593407126667897, 7744262079642179905375765602663, 30171043761093305994680851558869, 117597630210781663901746324225331, 458561947876935806749594717130925, 1788884368343729205856120971100515, 6981443969196495570407909952990701, 27257196946144743277974480215659227, 106459535529098331141662262340820946, 415958623937166895321232589472581905, 1625821580401401647402953931254356816, 6356938986158124620929149170311395591, 24864002912893732410596500679647960505, 97283130655514186614224683358416706858, 380753008219210969662359437609346308800, 1490680164030106846087610933390030116031, 5837906084538360412077215278231734442851, 22869547738174712268875737220918405203410, 89615344305142414341157078904090250606008, 351259549819278324846200875011727260239085, 1377182930383188109398552866582247022014891, 5400944877973007172570974785365420243950546, 21186513678773094338470847827890778260216630, 83130046054789815683947799692625357446941903, 326259041047634004574232654266686351332253935, 1280767931001720603177773021535078005822456816, 5028971012268165843635878009677687090186511226, 19750866543549132860571618211906105311702694125, 77587031736008360143111607391186701492095292083, 304849713771138184242179865086337124258333208820, 1198047147002902632689430827873677543554154018116, 4709245803128641791613979239398938668672109764571, 18514674646887293557083632333542521616887593163465, 72805820099466438292528945174047794903468838019658, 286351511602049060965633670926071067342254572932390, 1126456367790264882556299693640859123058483483341101, 4432094246726197026846072467575972990302997987396531, 17441409617328901293749904996294337716507065323873880, 68648409438502435187681949200049728455056372928924548, 270242751045902459662777290919953137309192683809709735, 1064022298592688055941905924709039250710929358790549291, 4190047849376648064610987086877761875331278603811303366, 16502788523718394002819341492099048383983006384506239160, 65007632890017595473423222409675240993717825573465901073, 256117142347383349976471444585491201982785261065038039295, 1009203788373811339853956374186817608376264309887951049226

Generating function in Maple syntax:
1/2*((x^4-2*x^3+4*x^2-3*x+1)*(1-4*x)^(1/2)-3*x^4+4*x^3-4*x^2+3*x-1)/x/(x-1)^3

Generating function in latex syntax:
\frac{\left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) \sqrt{1-4 x}-3 x^{4}+4 x^{3}-4 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}

Generating function in sympy syntax:
(-3*x**4/2 + 2*x**3 - 2*x**2 + 3*x/2 + sqrt(1 - 4*x)*(x**4 - 2*x**3 + 4*x**2 - 3*x + 1)/2 - 1/2)/(x*(x - 1)**3)

Implicit equation for the generating function in Maple syntax:
x*(x-1)^6*F(x)^2+(3*x^4-4*x^3+4*x^2-3*x+1)*(x-1)^3*F(x)+x^8-2*x^7+7*x^6-15*x^5+23*x^4-24*x^3+16*x^2-6*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{6} F \! \left(x \right)^{2}+\left(3 x^{4}-4 x^{3}+4 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}-2 x^{7}+7 x^{6}-15 x^{5}+23 x^{4}-24 x^{3}+16 x^{2}-6 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 58
a(6) = 177
a(7) = 554
a(8) = 1790
a(n+6) = -2*(-1+2*n)/(7+n)*a(n)+(3+13*n)/(7+n)*a(1+n)-3*(19+9*n)/(7+n)*a(n+2)+2*(56+17*n)/(7+n)*a(n+3)-(105+23*n)/(7+n)*a(n+4)+2*(23+4*n)/(7+n)*a(n+5)-2*n/(7+n), n >= 9

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) = 58
a \! \left(6\right) = 177
a \! \left(7\right) = 554
a \! \left(8\right) = 1790
a \! \left(n +6\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(3+13 n \right) a \! \left(1+n \right)}{7+n}-\frac{3 \left(19+9 n \right) a \! \left(n +2\right)}{7+n}+\frac{2 \left(56+17 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(105+23 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{7+n}-\frac{2 n}{7+n}, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21851/
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[5,x]+F[9,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]^2*F[8,x]
F[8,x] = x
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[5,x]*F[8,x]
F[11,x] = F[12,x]+F[18,x]
F[12,x] = F[13,x]*F[5,x]
F[13,x] = F[14,x]+F[5,x]
F[14,x] = F[15,x]*F[17,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[15,x]*F[8,x]
F[17,x] = F[16,x]
F[18,x] = F[15,x]*F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[8,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[8,x]
F[23,x] = F[15,x]+F[24,x]
F[24,x] = F[21,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_{5}\! \left(x \right)+F_{9}\! \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_{5} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{5}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{5}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{16}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \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_5(x) + F_9(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_5(x)**2*F_8(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_5(x)*F_8(x))
Eq(F_11(x), F_12(x) + F_18(x))
Eq(F_12(x), F_13(x)*F_5(x))
Eq(F_13(x), F_14(x) + F_5(x))
Eq(F_14(x), F_15(x)*F_17(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_15(x)*F_8(x))
Eq(F_17(x), F_16(x))
Eq(F_18(x), F_15(x)*F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_8(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_8(x))
Eq(F_23(x), F_15(x) + F_24(x))
Eq(F_24(x), F_21(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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 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": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 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}}, {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[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": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": 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