0132_0213_0231_0312_1032_1302_3021

Counting sequence:
1, 1, 2, 6, 17, 49, 150, 480, 1585, 5356, 18423, 64274, 226864, 808604, 2906166, 10520245, 38322622, 140373047, 516704222, 1910310677, 7090519487, 26411935367, 98702727917, 369949187717, 1390374138431, 5238459709589, 19782193477603, 74863493189233, 283873204841613, 1078395613987925, 4103721653335369, 15641399006286986, 59707320081299571, 228241277370280456, 873653604447681703, 3348341758779674022, 12847996626347362104, 49354680148278464584, 189794195543766306614, 730592995920403995738, 2815037523315165716696, 10856471385944162583250, 41905418239544231418300, 161886684453816520945350, 625885963204270900306730, 2421626665573341086307890, 9376328593329082188487070, 36329444852676292416696620, 140855080314703533018139670, 546464990151886427164393232, 2121373855924941434715167846, 8239990989377735985785255708, 32024409575454758431744014848, 124529190343402339601817093488, 484493679848587847017408132844, 1885922068609977371247920743916, 7344635393363975660952297413312, 28616744092330601901443284074092, 111549253553942062128701170071272, 435013698937935541103534336950628, 1697159632766670923686466428986140, 6623994114450636065088164842631276, 25863598328669072601302170196172116, 101023933133511738157057048107823481, 394748713199613309066041856353086178, 1543026357249027527072233715130727727, 6033612425718510899710444643049447614, 23600891116948590617256497787733950989, 92346832275237568605684020691696936239, 361454872797040382833519300285506122639, 1415209359268670758764498687776170346189, 5542657479027706254987453409986420984089, 21714133474000027139156855808644868321839, 85092365768104044381921876682999390977089, 333548497287613564252249931120909060973779, 1307809442306903138481172540868151751312709, 5129132997575563581140055079709768587714649, 20121229059506497113555462818523951585588449, 78953840098473099418671415171989447095934749, 309882807113082499888976845139533527984007919, 1216534813700239750162133052898664865894339709, 4776963266031766286263546528854061575944687179, 18761912972043826469472484932093408047483824029, 73705149642423493452611288715377356048169228879, 289608814092614209397569232067291465235419496699, 1138195212142644665697579579635858482227940247339, 4474151177699127264394728301535705056847670465759, 17591036631823688239601459680916660281231926876559, 69176266204594821829599253839716555166672190932859, 272085752386863879881583542923047475934199970183599, 1070374220707871877594971541034301895987856384288179, 4211578066044425090033951811514567893166660401521719, 16574167358034426711520367649773408014145155102518719, 65237087861970603755867012029830362966599070844220719, 256821737034415432676960097714028972667512529962774919, 1011211020440199166390402281735306501668192290169075569, 3982200253673953822926248492666038981886793259021493019, 15684627734392266492814408455356117606613161172196666269, 61786519199810652020865411664289724964799494612685542019, 243433451874139693159802864627668448285661124601225943209, 959251440087919576915888087608845296982474246909666355389

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

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

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

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

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

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 49
a(6) = 150
a(7) = 480
a(8) = 1585
a(n+6) = -2*(-5+2*n)/(7+n)*a(n)+(-13+9*n)/(7+n)*a(1+n)-2*(2+5*n)/(7+n)*a(n+2)+2*(12+5*n)/(7+n)*a(n+3)-2*(22+5*n)/(7+n)*a(n+4)+2*(17+3*n)/(7+n)*a(n+5), 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) = 17
a \! \left(5\right) = 49
a \! \left(6\right) = 150
a \! \left(7\right) = 480
a \! \left(8\right) = 1585
a \! \left(n +6\right) = -\frac{2 \left(-5+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(-13+9 n \right) a \! \left(1+n \right)}{7+n}-\frac{2 \left(2+5 n \right) a \! \left(n +2\right)}{7+n}+\frac{2 \left(12+5 n \right) a \! \left(n +3\right)}{7+n}-\frac{2 \left(22+5 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(17+3 n \right) a \! \left(n +5\right)}{7+n}, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22779/
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[9,x]^2*F[0,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[15,x]
F[14,x] = F[12,x]*F[9,x]
F[15,x] = F[16,x]
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[1,x]+F[18,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_{9} \left(x \right)^{2} F_{0}\! \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_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \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_0(x)*F_9(x)**2)
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) + F_15(x))
Eq(F_14(x), F_12(x)*F_9(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_1(x) + F_18(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": [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": [0, 3, 1, 2], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [0, 3, 1, 2], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [0, 3, 1, 2], "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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [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, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 2], [0, 0], [0, 2], [0, 2]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [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": 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