0132_1302_3021

Counting sequence:
1, 1, 2, 6, 21, 76, 275, 991, 3566, 12848, 46426, 168390, 613252, 2242584, 8233836, 30347562, 112259358, 416679413, 1551551202, 5794581045, 21701123969, 81482609105, 306686353431, 1156913954801, 4373414747778, 16565115485930, 62859156179586, 238942525168461, 909753626763413, 3469108699505571, 13247607876550752, 50657900380889436, 193961059039936108, 743548876301216007, 2853673938343681605, 10964108472702546551, 42168932632749774767, 162345429573889711010, 625597652082397803857, 2412895778375872275117, 9314349779564490129514, 35984903715562964723835, 139131782564648335477977, 538339105680019946514981, 2084468653208298218284180, 8076659876459884947285204, 31315049750976533342315383, 121492066174981387250616910, 471634433703751020059420320, 1831959014503898816059776907, 7119821643332319820191682679, 27685767421769799399595842108, 107713360204422373169266694725, 419276052650847557992048346372, 1632827700485365510055063414392, 6361846920770855023105146820239, 24798289633406746754571362658824, 96705092901946813551129626704197, 377276573314445348033326499948362, 1472472778777584973184926527664134, 5749181321844133324833408569590068, 22455909442739527393971472979802709, 87743724706705578185093196258474745, 342970966730738959347965946482132492, 1341065036277413386972527305390162969, 5245525919651106203246833829972409848, 20524409400930209073870595936388864298, 80332380661862502198371173807888061461, 314517648031635558554789130520335191414, 1231771225079845999401796180580174276477, 4825497978067686935205174222773208000840, 18909402725365767467370835710315792564330, 74119717436714548798136414857309662437562, 290607474741525133941399608086576846044015, 1139708598167679673205209630061643704568558, 4470868968635359729577993290581865843533578, 17542775234530752068316891869081577254992776, 68850986929040073900725191062221477265706836, 270286857840927967958032503957233047277284889, 1061304606454296282697646974335396671813965933, 4168243603258356908097322288806005037058615724, 16374261036132655547003992078353041696253007032, 64337419052891997397808611023494922087871478547, 252846345864784259215599666875152098803783207623, 993890956600370114939110562233212172947798563493, 3907579013851380279509107273238689113838633215338, 15366034385887274838940301900332880217296973639265, 60436448039506192560315070362918880965222519000788, 237748246439089293421579006619917161731732764643257, 935438394781750640848456431949892792692145385747912, 3681211896781538539507548310209762091167060566929533, 14489137429689611693547234595293031524685770977516283, 57038582203614336406753670139402299351719853386462058, 224578317420741093697138979045004514977226122600403696, 884378798629302156299301771068508146930126046544878150, 3483202716304266880838003549600834727519456486361115006, 13721060635522790489295852275422922236001866455843288274, 54058468983612026031414187614128982674514614302680287122, 213012743286434285322452865644278865026611342917081488049, 839483018312203887207579499543894461493695003115217067756, 3308883446647609136041872511526056325638842790825283405467

Generating function in Maple syntax:
1/2*(-4*(x^2-3*x+1)*(x^2-x+1/2)^2*(x-1)*(1-4*x)^(1/2)-6*x^8+48*x^7-140*x^6+210*x^5-194*x^4+115*x^3-44*x^2+10*x-1)/x^2/(x^6-8*x^5+23*x^4-29*x^3+20*x^2-7*x+1)/(x-1)

Generating function in latex syntax:
\frac{-4 \left(x^{2}-3 x +1\right) \left(x^{2}-x +\frac{1}{2}\right)^{2} \left(x -1\right) \sqrt{1-4 x}-6 x^{8}+48 x^{7}-140 x^{6}+210 x^{5}-194 x^{4}+115 x^{3}-44 x^{2}+10 x -1}{2 x^{2} \left(x^{6}-8 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1\right) \left(x -1\right)}

Generating function in sympy syntax:
(-3*x**8 + 24*x**7 - 70*x**6 + 105*x**5 - 97*x**4 + 115*x**3/2 - 22*x**2 + 5*x + sqrt(1 - 4*x)*(x - 1)*(-4*x**2 + 12*x - 4)*(x**2 - x + 1/2)**2/2 - 1/2)/(x**2*(x - 1)*(x**6 - 8*x**5 + 23*x**4 - 29*x**3 + 20*x**2 - 7*x + 1))

Implicit equation for the generating function in Maple syntax:
x^2*(x^6-8*x^5+23*x^4-29*x^3+20*x^2-7*x+1)*(x-1)^2*F(x)^2+(x-1)*(6*x^8-48*x^7+140*x^6-210*x^5+194*x^4-115*x^3+44*x^2-10*x+1)*F(x)+9*x^8-56*x^7+145*x^6-209*x^5+191*x^4-114*x^3+44*x^2-10*x+1 = 0

Implicit equation for the generating function in latex syntax:
x^{2} \left(x^{6}-8 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)^{2}+\left(x -1\right) \left(6 x^{8}-48 x^{7}+140 x^{6}-210 x^{5}+194 x^{4}-115 x^{3}+44 x^{2}-10 x +1\right) F \! \left(x \right)+9 x^{8}-56 x^{7}+145 x^{6}-209 x^{5}+191 x^{4}-114 x^{3}+44 x^{2}-10 x +1 = 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) = 275
a(7) = 991
a(8) = 3566
a(9) = 12848
a(10) = 46426
a(11) = 168390
a(12) = 613252
a(n+11) = 4*(2*n+3)/(13+n)*a(n)+4*(125*n+389)/(13+n)*a(2+n)-2*(49*n+111)/(13+n)*a(n+1)-5*(279*n+1135)/(13+n)*a(n+3)+(12273+2399*n)/(13+n)*a(n+4)-4*(688*n+4279)/(13+n)*a(n+5)+2*(1091*n+8026)/(13+n)*a(n+6)-18*(67*n+570)/(13+n)*a(n+7)+8*(57*n+550)/(13+n)*a(n+8)-2*(56*n+603)/(13+n)*a(n+9)+2*(8*n+95)/(13+n)*a(n+10)+3*(n+3)/(13+n), n >= 13

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) = 275
a \! \left(7\right) = 991
a \! \left(8\right) = 3566
a \! \left(9\right) = 12848
a \! \left(10\right) = 46426
a \! \left(11\right) = 168390
a \! \left(12\right) = 613252
a \! \left(n +11\right) = \frac{4 \left(2 n +3\right) a \! \left(n \right)}{13+n}+\frac{4 \left(125 n +389\right) a \! \left(2+n \right)}{13+n}-\frac{2 \left(49 n +111\right) a \! \left(n +1\right)}{13+n}-\frac{5 \left(279 n +1135\right) a \! \left(n +3\right)}{13+n}+\frac{\left(12273+2399 n \right) a \! \left(n +4\right)}{13+n}-\frac{4 \left(688 n +4279\right) a \! \left(n +5\right)}{13+n}+\frac{2 \left(1091 n +8026\right) a \! \left(n +6\right)}{13+n}-\frac{18 \left(67 n +570\right) a \! \left(n +7\right)}{13+n}+\frac{8 \left(57 n +550\right) a \! \left(n +8\right)}{13+n}-\frac{2 \left(56 n +603\right) a \! \left(n +9\right)}{13+n}+\frac{2 \left(8 n +95\right) a \! \left(n +10\right)}{13+n}+\frac{3 n +9}{13+n}, \quad n \geq 13

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/298/
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[9,x]
F[4,x] = F[10,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[8,x] = F[5,x]^2*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[5,x]*F[9,x]
F[12,x] = F[13,x]+F[15,x]
F[13,x] = 1/F[9,x]*F[14,x]
F[14,x] = F[2,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[23,x]*F[9,x]
F[17,x] = F[18,x]+F[26,x]
F[18,x] = F[5,x]^2*F[19,x]
F[19,x] = F[20,x]+F[5,x]
F[20,x] = F[21,x]*F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[9,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[24,x]*F[9,x]
F[26,x] = F[24,x]^2*F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[9,x]
F[29,x] = F[10,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[23,x]*F[5,x]*F[7,x]*F[9,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_{9}\! \left(x \right)
F_{4}\! \left(x \right) = F_{10}\! \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_{8}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{15}\! \left(x \right)
F_{13}\! \left(x \right) = \frac{F_{14}\! \left(x \right)}{F_{9}\! \left(x \right)}
F_{14}\! \left(x \right) = F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{18}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{5}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{24}\! \left(x \right) F_{9}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24} \left(x \right)^{2} F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{9}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right) F_{5}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \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_9(x))
Eq(F_4(x), F_10(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))
Eq(F_8(x), F_5(x)**2*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_5(x)*F_9(x))
Eq(F_12(x), F_13(x) + F_15(x))
Eq(F_13(x), F_14(x)/F_9(x))
Eq(F_14(x), F_2(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_23(x)*F_9(x))
Eq(F_17(x), F_18(x) + F_26(x))
Eq(F_18(x), F_19(x)*F_5(x)**2)
Eq(F_19(x), F_20(x) + F_5(x))
Eq(F_20(x), F_21(x)*F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_9(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_24(x)*F_9(x))
Eq(F_26(x), F_24(x)**2*F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_9(x))
Eq(F_29(x), F_10(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_23(x)*F_5(x)*F_7(x)*F_9(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, 1, 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": [{"class_module": "comb_spec_searcher.strategies.rule", "idx": 0, "original_rule": {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 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, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": 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