0132_0213_0231_0312_1023_1230

Counting sequence:
1, 1, 2, 6, 18, 54, 167, 532, 1742, 5838, 19943, 69201, 243232, 864117, 3097675, 11190209, 40693609, 148845378, 547227742, 2021055280, 7494785042, 27895625622, 104173970474, 390210766526, 1465695537694, 5519431121988, 20833592944032, 78809126057196, 298719066237036, 1134389810267392, 4315386712861368, 16443174012013725, 62750231779347750, 239810457828648691, 917713237384711004, 3516399212847978391, 13489961471598084735, 51810310665357082762, 199199616094247427795, 766661234088434063039, 2953513325053819541628, 11388699971290666331267, 43953149988341007498505, 169773026295089073640052, 656286735262979320699917, 2538921925954224218464889, 9829272997878214207618106, 38079940089083308615554205, 147625455426017089487988441, 572670071164081344070205638, 2222873712923832996013789453, 8633395695088239437169028691, 33550202928449999136149028072, 130450564791304207527353259587, 507487399962439560245788767359, 1975262030280381793268154811006, 7691949612934658414321676012585, 29967667290172972085733002781367, 116806538952063901500300256569340, 455483198008536428780728042370051, 1776896659285918239141402855523839, 6934745311572312579755937284883410, 27075196750446578979852361282831337, 105749905842341364564518997746860456, 413190549868417478885932097644074954, 1615019596728907479669643405467527122, 6314769008565490004464936699663261951, 24699311212566418871978226056052140873, 96639697771286596808142740699603426724, 378238269079669018245713035252862228897, 1480848297782978427450372712301693525471, 5799453307087431020351269405023093034618, 22719107572443919726868484285039420494139, 89026581963319610995988934931347370932157, 348954648406435579578604413296841722285386, 1368156896878740081128746777285596731007023, 5365588310583178138048184914601690052410409, 21047975882219602204197443054098776649670192, 82587060731955733038553179266268212878957181, 324130283108271080148672702703605292014708523, 1272419983339108070239786274224137739490137454, 4996225982099895819379757549283856087576222377, 19622391012242725141345726889947007047227331601, 77082832689453375105810028832988522098060139738, 302870523658299344050373408274941581606473200419, 1190276193246248213935312772380153651643361698517, 4678727528671578651235233402817194880098468014836, 18394795934066124045753637771078495783101340074413, 72334822916757464106389400357322462441236856359829, 284500596642617995040114537249267331306632784514162, 1119181176890394871349018892325052897167150949193131, 4403492666902425684248084389287274055348035859041293, 17328943591004706073831530113406162321841633628072824, 68206089571556012884343250370281283901507928356667317, 268502814378881313061045847099097869847434692245555541, 1057176714232280418036207518264620370811906482476732908, 4163109811575028324559980599868854952423586842773516699, 16396766023049006799218111200034738148245222406259540007, 64590278346073245676801520431744356269813816681124816656, 254473960298156198741215341960804338699789496761547487303, 1002733284296261166880384269917617506196112107330776921039

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

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

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

Implicit equation for the generating function in Maple syntax:
x*(x^2+x-1)^2*F(x)^2-(x^2+x-1)*(2*x^4+x^3+x-1)*F(x)+x^7+2*x^6-x^4+2*x^3-2*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x^{2}+x -1\right)^{2} F \! \left(x \right)^{2}-\left(x^{2}+x -1\right) \left(2 x^{4}+x^{3}+x -1\right) F \! \left(x \right)+x^{7}+2 x^{6}-x^{4}+2 x^{3}-2 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 54
a(6) = 167
a(7) = 532
a(n+6) = -2*(-1+2*n)/(7+n)*a(n)-3*(n-1)/(7+n)*a(1+n)+(13+9*n)/(7+n)*a(n+2)-2*a(n+3)-2*(17+4*n)/(7+n)*a(n+4)+2*(17+3*n)/(7+n)*a(n+5), n >= 8

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) = 18
a \! \left(5\right) = 54
a \! \left(6\right) = 167
a \! \left(7\right) = 532
a \! \left(n +6\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{7+n}-\frac{3 \left(n -1\right) a \! \left(1+n \right)}{7+n}+\frac{\left(13+9 n \right) a \! \left(n +2\right)}{7+n}-2 a \! \left(n +3\right)-\frac{2 \left(17+4 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 8

Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22123/
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[17,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[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[8,x]
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,x]
F[16,x] = F[6,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[5,x]*F[8,x]
F[19,x] = F[20,x,1]
F[20,x,k[0]] = (F[21,x,k[0]]*k[0]-F[21,x,1])/(-1+k[0])
F[21,x,k[0]] = F[22,x,k[0]]+F[26,x,k[0]]
F[22,x,k[0]] = F[1,x]+F[23,x,k[0]]
F[23,x,k[0]] = F[24,x,k[0]]
F[24,x,k[0]] = F[22,x,k[0]]*F[25,x,k[0]]
F[25,x,k[0]] = k[0]*x
F[26,x,k[0]] = F[27,x,k[0]]
F[27,x,k[0]] = F[20,x,k[0]]*F[22,x,k[0]]*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_{17}\! \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_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \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_{6}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x , 1\right)
F_{20}\! \left(x , y\right) = \frac{y F_{21}\! \left(x , y\right)-F_{21}\! \left(x , 1\right)}{-1+y}
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)+F_{26}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{22}\! \left(x , y\right) F_{25}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = y x
F_{26}\! \left(x , y\right) = F_{27}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{20}\! \left(x , y\right) F_{22}\! \left(x , y\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_17(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)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(x))
Eq(F_16(x), F_6(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_5(x)*F_8(x))
Eq(F_19(x), F_20(x, 1))
Eq(F_20(x, k_0), (-k_0*F_21(x, k_0) + F_21(x, 1))/(1 - k_0))
Eq(F_21(x, k_0), F_22(x, k_0) + F_26(x, k_0))
Eq(F_22(x, k_0), F_1(x) + F_23(x, k_0))
Eq(F_23(x, k_0), F_24(x, k_0))
Eq(F_24(x, k_0), F_22(x, k_0)*F_25(x, k_0))
Eq(F_25(x, k_0), k_0*x)
Eq(F_26(x, k_0), F_27(x, k_0))
Eq(F_27(x, k_0), F_20(x, k_0)*F_22(x, k_0)*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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 1, 2], "pos": [[0, 1], [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, 0]]}, {"patt": [1, 2, 3, 0], "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, 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, 1, 2], "pos": [[0, 2], [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, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [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}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 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