0132_0213_0231_1302_3021

Counting sequence:
1, 1, 2, 6, 19, 59, 182, 570, 1833, 6060, 20520, 70825, 248145, 879814, 3149762, 11367590, 41308905, 151009399, 554922244, 2048661047, 7594582245, 28258772115, 105502992394, 395099300099, 1483758149591, 5586440522157, 21083093905716, 79741183182753, 302211426860421, 1147511530279273, 4364813250239646, 16629786833204998, 63456319315768349, 242487428550561680, 927881250301228140, 3555087678380882393, 13637406720039688829, 52373091766397193814, 201350749555121234822, 774894603847740023499, 2985066042481268494843, 11509763714186306904014, 44418180466962916992932, 171561212709274998013647, 663169838582574930900469, 2565441942672660889760788, 9931545956445690219263234, 38474693921024185999550547, 149150395305983564613105447, 578565609385208950669541296, 2245683603756916367381316944, 8721711333584872522870206647, 33892381845690134377949883009, 131777218342187791532455086538, 512634215800607633332098032662, 1995241569222868667533347639641, 7769554233318785487952782529383, 30269269930654889173466931992112, 117979325195165668890705319002100, 460045982812054562360343669883103, 1794657365319078744368922598130837, 7003912758183000501069222301737066, 27344689517851531638901201885607362, 106800389495261856703875230802316486, 417287131525678514574387224510197505, 1631001839420903739598027007745931047, 6377147027956595443728925899075249276, 24942866440101531726901286165989170031, 97591024892602827671960237987073484301, 381955535082368935853955986087203548875, 1495378570700390510618875654640363261442, 5856269945046254350702326221100058629391, 22941348678422085777360601856786532209911, 89896174926901481721005519767539989962841, 352358309202701861380211413493995477852860, 1381483246195229733621308665108415869713057, 5417780676392067451828262587588412225197741, 21252446074878730188782586957259610871140381, 83388326841927817747849250290017492445217126, 327271109575727500541911953581926503871130175, 1284734809831700436598899944648946106581549267, 5044523716102333877830932355710313149832825251, 19811859221957971863207101165296056966083875296, 77826286917042066244659731673545631331478961781, 305788472442501931835372838802479260594671356465, 1201731428189903910701192487388624357877750615107, 4723708693944178723987888290635425008835120419026, 18571462887547471679614657492498911362740421354501, 73028848524423350747571333218832749614003120303491, 287227620591030183620129589160192714822023850222775, 1129898676971195347837752266551268533816326304313852, 4445622235048756148477060472661235085010466659803841, 17494584437899187993545152009025394152141793504939261, 68857467179168730909918820231356043806364757160603831, 271064826887559229170719745857001540715776430890717670, 1067255561410076302099733917565770477741607946866310347, 4202766813971939333069214419367653917493863314053975531, 16552831408643682271421944140598894831956314232242549541, 65204562494110237567959101088805980686243649037084583896, 256892239395284042575843340524741395729674648913478065505, 1012255025313941547698033475000998378555234797938936530537

Generating function in Maple syntax:
1/2*((-3*x^4+6*x^3-7*x^2+4*x-1)*(1-4*x)^(1/2)-2*x^5+7*x^4-8*x^3+7*x^2-4*x+1)/x/(x-1)^4

Generating function in latex syntax:
\frac{\left(-3 x^{4}+6 x^{3}-7 x^{2}+4 x -1\right) \sqrt{1-4 x}-2 x^{5}+7 x^{4}-8 x^{3}+7 x^{2}-4 x +1}{2 x \left(x -1\right)^{4}}

Generating function in sympy syntax:
(-x**5 + 7*x**4/2 - 4*x**3 + 7*x**2/2 - 2*x + sqrt(1 - 4*x)*(-3*x**4 + 6*x**3 - 7*x**2 + 4*x - 1)/2 + 1/2)/(x*(x - 1)**4)

Implicit equation for the generating function in Maple syntax:
x*(x-1)^8*F(x)^2+(2*x^5-7*x^4+8*x^3-7*x^2+4*x-1)*(x-1)^4*F(x)+x^9+2*x^8-18*x^7+52*x^6-83*x^5+87*x^4-62*x^3+29*x^2-8*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{8} F \! \left(x \right)^{2}+\left(2 x^{5}-7 x^{4}+8 x^{3}-7 x^{2}+4 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{9}+2 x^{8}-18 x^{7}+52 x^{6}-83 x^{5}+87 x^{4}-62 x^{3}+29 x^{2}-8 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 59
a(6) = 182
a(7) = 570
a(8) = 1833
a(9) = 6060
a(n+6) = -6*(2*n+1)/(n+7)*a(n)+3*(13*n+12)/(n+7)*a(n+1)-(61*n+135)/(n+7)*a(n+2)+(194+57*n)/(n+7)*a(n+3)-(31*n+143)/(n+7)*a(n+4)+(52+9*n)/(n+7)*a(n+5)+1/2*(n^2-13*n+10)/(n+7), 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) = 19
a \! \left(5\right) = 59
a \! \left(6\right) = 182
a \! \left(7\right) = 570
a \! \left(8\right) = 1833
a \! \left(9\right) = 6060
a \! \left(n +6\right) = -\frac{6 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{3 \left(13 n +12\right) a \! \left(n +1\right)}{n +7}-\frac{\left(61 n +135\right) a \! \left(n +2\right)}{n +7}+\frac{\left(194+57 n \right) a \! \left(n +3\right)}{n +7}-\frac{\left(31 n +143\right) a \! \left(n +4\right)}{n +7}+\frac{\left(52+9 n \right) a \! \left(n +5\right)}{n +7}+\frac{n^{2}-13 n +10}{2 n +14}, \quad n \geq 10

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21581/
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[14,x] = F[15,x]*F[18,x]*F[23,x]
F[15,x] = F[16,x]+F[20,x]
F[16,x] = F[17,x]^2
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[8,x]
F[20,x] = F[21,x]
F[21,x] = F[17,x]*F[22,x]*F[5,x]*F[8,x]
F[22,x] = F[6,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[23,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_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{23}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17} \left(x \right)^{2}
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{6}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{23}\! \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))
Eq(F_14(x), F_15(x)*F_18(x)*F_23(x))
Eq(F_15(x), F_16(x) + F_20(x))
Eq(F_16(x), F_17(x)**2)
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_8(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_17(x)*F_22(x)*F_5(x)*F_8(x))
Eq(F_22(x), F_6(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_23(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, 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": [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": [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": [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"}}, {"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, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 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, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 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], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], 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