0213_1032_1320_3021

Counting sequence:
1, 1, 2, 6, 20, 64, 199, 620, 1972, 6443, 21598, 73981, 257824, 910851, 3253088, 11721696, 42548766, 155419210, 570789577, 2106261215, 7805140520, 29032878384, 108362718021, 405708054109, 1523259229138, 5734008190700, 21636029976541, 81818706133301, 310036818327700, 1177055612106322, 4476593262801667, 17053547352436572, 65065778700531812, 248610781279872147, 951215830546166718, 3644144753876932385, 13977779494892366096, 53675727166702258615, 206342345648162466140, 794044837255214939193, 3058618775966548925396, 11792569250860921685941, 45506653196579953319842, 175754607407320519851227, 679339914917902648716752, 2627849557937220319845129, 10172605684938002170724082, 39406567215236566610593189, 152755508111502176131131856, 592522745144110070880733427, 2299756019741926274089062416, 8931337111627225134344409327, 34705570340874990397513943112, 134933707350251446300977168595, 524893733912940074855927754076, 2042883431431258695563903698511, 7954796878579660148136635400608, 30989914897005706863891933115551, 120784252300277840297197116380688, 470968777847860704226428916277203, 1837212359052857820433771874006248, 7169781580932950494350252030179711, 27991489174036102073472909213922112, 109323633337740715004325055646043576, 427134655285145162207389837145674566, 1669449283031761428750003110268980722, 6527314210629637905748397702468706673, 25529604829082215876813701681371057327, 99884377836570559967605814871368558000, 390922582249695545711685391634214265824, 1530451789931976142443814215130547541813, 5993498686310574714761629433530955271745, 23478446818327487554035441853590093711826, 91998971972096032880462607812042047641968, 360593475997678892201432351630389054125785, 1413744045462745025437361655962743098991725, 5544196671318797815144013858105839163329404, 21747953028696222028734542941441407736879534, 85331068615720659233768690500857116773389439, 334890055749935867895014684841720019292929757, 1314622080572203370844878668883452956257449234, 5161793540318361974109299657108954309628832382, 20272107531254307253306244168359562882004089335, 79633050761315849070040279459486631499098081461, 312882800884036308185768242415628889143886459456, 1229593887219008168484418886685155257471352788992, 4833160637580734933968264703133827895797336911193, 19001515026614769424025877008303104757708315265087, 74718939727391570720032224403503776084841749078430, 293870990551728374688212757909645809358875605457894, 1156017520896363804371023888081820422726254590118803, 4548330334574478471123438716445476850429669428468485, 17898544798223366057013867359341245426869321920828796, 70446578313243438051861908165750621165396950670837448, 277317265848074005313291102536951443100046164174852865, 1091860517144782033785881920892873469955830772503255145, 4299610648214004890399887841112781971205239878972003734, 16934069248359090681722082800815771708058392746989151224, 66705605338439060213292410182942688399964036578277904049, 262803249723991537162870949443916782645176141003101777817, 1035535960034735824923903074135514680085041488897254179064

Generating function in Maple syntax:
1/2*((-2*x^4+9*x^3-10*x^2+5*x-1)*(1-4*x)^(1/2)+2*x^5+6*x^4-11*x^3+10*x^2-5*x+1)/x/(2*x-1)/(x-1)^3

Generating function in latex syntax:
\frac{\left(-2 x^{4}+9 x^{3}-10 x^{2}+5 x -1\right) \sqrt{1-4 x}+2 x^{5}+6 x^{4}-11 x^{3}+10 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(x**5 + 3*x**4 - 11*x**3/2 + 5*x**2 - 5*x/2 + sqrt(1 - 4*x)*(-2*x**4 + 9*x**3 - 10*x**2 + 5*x - 1)/2 + 1/2)/(x*(x - 1)**3*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^6*F(x)^2-(2*x-1)*(2*x^5+6*x^4-11*x^3+10*x^2-5*x+1)*(x-1)^3*F(x)+x^9+10*x^8-39*x^7+107*x^6-175*x^5+175*x^4-111*x^3+44*x^2-10*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)^{2}-\left(2 x -1\right) \left(2 x^{5}+6 x^{4}-11 x^{3}+10 x^{2}-5 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}+10 x^{8}-39 x^{7}+107 x^{6}-175 x^{5}+175 x^{4}-111 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) = 20
a(5) = 64
a(6) = 199
a(7) = 620
a(8) = 1972
a(9) = 6443
a(n+7) = 8*(1+2*n)/(n+8)*a(n)-4*(42+25*n)/(n+8)*a(n+1)+110*(5+2*n)/(n+8)*a(n+2)-5*(170+49*n)/(n+8)*a(n+3)+(716+157*n)/(n+8)*a(n+4)-(336+59*n)/(n+8)*a(n+5)+2*(41+6*n)/(n+8)*a(n+6)+(3*n-10)/(n+8), n >= 10

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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 199
a \! \left(7\right) = 620
a \! \left(8\right) = 1972
a \! \left(9\right) = 6443
a \! \left(n +7\right) = \frac{8 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{4 \left(42+25 n \right) a \! \left(n +1\right)}{n +8}+\frac{110 \left(5+2 n \right) a \! \left(n +2\right)}{n +8}-\frac{5 \left(170+49 n \right) a \! \left(n +3\right)}{n +8}+\frac{\left(716+157 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(336+59 n \right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(41+6 n \right) a \! \left(n +6\right)}{n +8}+\frac{3 n -10}{n +8}, \quad n \geq 10

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21383/
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[8,x]
F[11,x] = F[12,x]+F[24,x]
F[12,x] = F[13,x]+F[4,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[20,x]
F[15,x] = F[16,x]+F[23,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[20,x]*F[8,x]
F[19,x] = F[1,x]+F[17,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[20,x]*F[8,x]
F[23,x] = F[21,x]*F[6,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[5,x]*F[6,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[20,x]^2
F[28,x] = F[29,x]
F[29,x] = F[20,x]*F[5,x]*F[6,x]*F[8,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_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{5}\! \left(x \right) F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20} \left(x \right)^{2}
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right) F_{5}\! \left(x \right) F_{6}\! \left(x \right) F_{8}\! \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_8(x))
Eq(F_11(x), F_12(x) + F_24(x))
Eq(F_12(x), F_13(x) + F_4(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_20(x))
Eq(F_15(x), F_16(x) + F_23(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_20(x)*F_8(x))
Eq(F_19(x), F_1(x) + F_17(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_20(x)*F_8(x))
Eq(F_23(x), F_21(x)*F_6(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_5(x)*F_6(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_20(x)**2)
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_20(x)*F_5(x)*F_6(x)*F_8(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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"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, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 2], [0, 0], [0, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], 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