0213_1032_1302_1320_2031

Counting sequence:
1, 1, 2, 6, 19, 60, 190, 608, 1974, 6512, 21828, 74288, 256404, 896288, 3168712, 11314785, 40756630, 147929582, 540496813, 1986313104, 7336817206, 27221351321, 101398426804, 379039546308, 1421381776621, 5345335614812, 20154038954712, 76168119369425, 288484817648616, 1094805445385420, 4162460072804913, 15852762224789753, 60471480781834558, 231015749361942087, 883765616554170985, 3385324797508100796, 12983685048007832831, 49853959379362232569, 191636261154887245660, 737406030030543849441, 2840294041386604360553, 10950300216030962190408, 42254694443066795027301, 163189304873549104067945, 630752664358233260635184, 2439839424907406128840617, 9444594120666408446273277, 36585701774761588149624914, 141818388100825996434706385, 550091086664627905274958093, 2135040766036338786827939514, 8291565125795811845600784689, 32219259420442523164080589505, 125266164299372380051654599234, 487284084629429477641860619557, 1896498171060805635729161892593, 7384759843546288911771036689498, 28769114386401995979825866366481, 112128392915886024314664549619197, 437216839359669579464756821469810, 1705547725403837532478387234587701, 6655956017803241259031749706619961, 25985479776047968880188432983162402, 101489054179311643104142994490909686, 396524972243769185784963332850728795, 1549814447916619089711665049597636888, 6059570928330611956230942161206413558, 23700223870218685850734921146921666425, 92727176464844381480454674023328537456, 362912087909034296980585394923315888182, 1420795664022237859018807370771670549129, 5564084981923499009319649236471129217994, 21796368484320185605396959358756446902342, 85408137178433184870496051995223908933621, 334761642168006905294135355044771599935150, 1312472487088890105943402678019069881128518, 5147065342748167936031053830330669507676493, 20190222687883302402857599297831105978443090, 79219410293375563777407836119897422074628426, 310905493401091817350600684845847564972467827, 1220474782811154824648887094506925382137188114, 4792148630833662585524398648369635116620277474, 18820464012943962853461353736206479231972896347, 73930997581518725195197162598884859174946551546, 290480309475874158488498451321239755282830291430, 1141559374886179838716685521651102845275866963043, 4487142325247751308845064493072013108366116055998, 17641221463050583241904522617258042952188825418990, 69370197467424813691233358974173488943492883353531, 272835421642355519336403880598042262882873940356522, 1073273128090176659138158646478493140576710796319354, 4222791486857298116694322756228525640389217115531859, 16617556148068131142735569994469284912500547984750142, 65405025952642385627923817616211204232032165389383286, 257471941371837215849026143398467703284973323103884435, 1013729139871329302209937891407846581504251542754492388, 3991955214519160305900680975707121881494852018383811598, 15722427970378631956783817643555373413471226419945755375, 61933033626016089482118155949732564635416687759674635296, 244001493777411210688374283550561446436087273283412955022, 961454324767578645950451038970387542548907286146507198239

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

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

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

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

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

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = 4*(3+2*n)/(n+6)*a(n)-2*(36+19*n)/(n+6)*a(1+n)+3*(50+19*n)/(n+6)*a(n+2)-12*(11+3*n)/(n+6)*a(n+3)+2*(24+5*n)/(n+6)*a(n+4), n >= 5

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(n +5\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{n +6}-\frac{2 \left(36+19 n \right) a \! \left(1+n \right)}{n +6}+\frac{3 \left(50+19 n \right) a \! \left(n +2\right)}{n +6}-\frac{12 \left(11+3 n \right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(24+5 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 5

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21945/
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[13,x]
F[12,x] = F[4,x]*F[5,x]
F[13,x] = F[14,x]*F[16,x]*F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[25,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[8,x]
F[21,x] = F[16,x]+F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[22,x]*F[8,x]
F[25,x] = F[14,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[30,x]
F[27,x] = 0
F[28,x] = F[29,x]*F[8,x]
F[29,x] = F[19,x]+F[26,x]
F[30,x] = F[31,x]*F[8,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[14,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[32,x]*F[8,x]
F[35,x] = F[26,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,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_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \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_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{14}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \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_13(x))
Eq(F_12(x), F_4(x)*F_5(x))
Eq(F_13(x), F_14(x)*F_16(x)*F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_25(x))
Eq(F_18(x), F_1(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_16(x) + F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_22(x)*F_8(x))
Eq(F_25(x), F_14(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_30(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_29(x)*F_8(x))
Eq(F_29(x), F_19(x) + F_26(x))
Eq(F_30(x), F_31(x)*F_8(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_14(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_32(x)*F_8(x))
Eq(F_35(x), F_26(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(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, 0, 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": [2, 0, 3, 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, 0, 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": [2, 0, 3, 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, 0, 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": [2, 0, 3, 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": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 1], [0, 1]]}], "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": [1, 0, 2], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}], "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": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], 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