0213_0321_1302

Counting sequence:
1, 1, 2, 6, 21, 76, 277, 1015, 3743, 13893, 51874, 194693, 733983, 2777748, 10547615, 40169157, 153377405, 586999345, 2251219778, 8649959375, 33292809573, 128339862690, 495439282381, 1915080834953, 7411525121832, 28715269021585, 111370029984771, 432357420153744, 1679998103348673, 6533424431349207, 25428249702334132, 99040992545704793, 386026992795174410, 1505592412085143310, 5875820770079204456, 22944925458070867926, 89649630983363227170, 350462768743485418742, 1370741304617955577938, 5363877818775217957622, 20999178432505189595346, 82246371110128321433568, 322265474978070129033940, 1263239083325703234148944, 4953643141601053763066782, 19432308290922938420728864, 76256721234605870591364982, 299350417956474933452846608, 1175503476478971876041352158, 4617478821913038756180295628, 18143363258821179676623684668, 71311208538427974267336715794, 280362810412064190971708408278, 1102557769633807524569159648050, 4337071722284807162411659976330, 17064844096404357072169960740616, 67160648619311487706515768382568, 264381353940919730573965510624986, 1040990735460504171155649137752134, 4099773696676216340570437908296094, 16149791093818573661862966556682606, 63630465661410690398110689113846976, 250756245156580611096187802835924288, 988380791237734471471684978830741710, 3896549968719365818039526129840329781, 15364458540747460244221228516354844388, 60594480964975655623214380380361944665, 239015199300424743907888143830862334807, 942958410226765060980474149512104007490, 3720764157529219582980589327302834580775, 14683934195204250863474629146078377148333, 57959091765248207085012522529729860782640, 228806213243190126827541062842118904955783, 903398718467325715517364878223041457647345, 3567425562177361370474004240351146205183216, 14089403382090650826568379555549603930904373, 55653312531862264044624622608332993970560217, 219861264608666874360533628487667865255275174, 868688992211386454364635564759743182320181559, 3432704871487513495826981412139849685453131489, 13566376667106569360720695598031822953582070266, 53622280834382391036102140220234157679618160085, 211972487342791482010739476457086235820199273171, 838041153090405762908964418795744257267752293784, 3313612401782453395551911077263815871569914663291, 13103505404262793777114517139952438609651932065479, 51822892523379996259616013423997297097155411793642, 204976070153290295651505524448144219651486963408109, 810832310089563663630430609742252974551769958184343, 3207778320036197765122281125033497611810841199797484, 12691768585341623304407974512102562883982607787732975, 50220790343208486274841983396791433459123908788073435, 198741094088841604597570133172525366258183326360834914, 786563346591948061261780554378442559705173636283263353, 3113298867219132502877192726913385854335565795650375007, 12323901346004857192220845385126644212426573860230915556, 48788238286749818390153127475053923696561545837639726149, 193161619936937235298091586482165533129650532192828556071, 764829442999277863877873893344405832047590919471157242216, 3028626459900916561221245886227103802709931431821084263959, 11993984932587064763526121740945703996098724264245637834603

Generating function in Maple syntax:
-(-3*x^2+(x^2+x-1)*(1-4*x)^(1/2)+5*x-1)*(x-1)/(2*x^4+8*x^3-18*x^2+12*x-2)

Generating function in latex syntax:
-\frac{\left(-3 x^{2}+\left(x^{2}+x -1\right) \sqrt{1-4 x}+5 x -1\right) \left(x -1\right)}{2 x^{4}+8 x^{3}-18 x^{2}+12 x -2}

Generating function in sympy syntax:
(x - 1)*(3*x**2 - 5*x - sqrt(1 - 4*x)*(x**2 + x - 1) + 1)/(2*x**4 + 8*x**3 - 18*x**2 + 12*x - 2)

Implicit equation for the generating function in Maple syntax:
(x^4+4*x^3-9*x^2+6*x-1)*F(x)^2-(x-1)*(3*x^2-5*x+1)*F(x)+x*(x-1)^2 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{4}+4 x^{3}-9 x^{2}+6 x -1\right) F \! \left(x \right)^{2}-\left(x -1\right) \left(3 x^{2}-5 x +1\right) F \! \left(x \right)+x \left(x -1\right)^{2} = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 76
a(6) = 277
a(7) = 1015
a(n+8) = -2*(1+2*n)/(n+8)*a(n)-3*(2+5*n)/(n+8)*a(n+1)+2*(31+24*n)/(n+8)*a(n+2)-(-76+7*n)/(n+8)*a(n+3)-(454+85*n)/(n+8)*a(n+4)+21*(28+5*n)/(n+8)*a(n+5)-(336+53*n)/(n+8)*a(n+6)+2*(43+6*n)/(n+8)*a(n+7), n >= 8

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) = 76
a \! \left(6\right) = 277
a \! \left(7\right) = 1015
a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{3 \left(2+5 n \right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(31+24 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(-76+7 n \right) a \! \left(n +3\right)}{n +8}-\frac{\left(454+85 n \right) a \! \left(n +4\right)}{n +8}+\frac{21 \left(28+5 n \right) a \! \left(n +5\right)}{n +8}-\frac{\left(336+53 n \right) a \! \left(n +6\right)}{n +8}+\frac{2 \left(43+6 n \right) a \! \left(n +7\right)}{n +8}, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/318/
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[7,x] = F[13,x]+F[8,x]
F[8,x] = F[4,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[14,x] = F[0,x]*F[15,x]*F[9,x]
F[15,x] = F[16,x]
F[16,x] = F[9,x]^2*F[12,x]*F[17,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[12,x]*F[17,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_{7}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \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_{14}\! \left(x \right) = F_{0}\! \left(x \right) F_{15}\! \left(x \right) F_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \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))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_4(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))
Eq(F_14(x), F_0(x)*F_15(x)*F_9(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_12(x)*F_17(x)*F_9(x)**2)
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_12(x)*F_17(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "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], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "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], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "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], "pos": [[3, 1]]}, {"patt": [1, 0], "pos": [[2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [2, 1], [3, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [3, 0], [3, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [3, 0], [3, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": 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