0132_0231_0312_1023_1230

Counting sequence:
1, 1, 2, 6, 19, 58, 181, 578, 1888, 6297, 21393, 73843, 258354, 914304, 3267250, 11772591, 42722490, 156000815, 572728825, 2112751055, 7827021970, 29107327220, 108618428950, 406594468520, 1526358567304, 5744931151682, 21674803769946, 81957243246718, 310534731738068, 1178854734213488, 4483125686810756, 17077371026149537, 65153015617886766, 248931412218588091, 952398320176603529, 3648519610211111191, 13994012724152999026, 53736126407851622348, 206567646792084485358, 794887256955706262900, 3061775681453721024112, 11804424300184716724030, 45551259880217542347070, 175922759403895962878810, 679974901451883765508700, 2630251408320374115580960, 10181704988905603075490300, 39441090806401065671998510, 152886678786606827819341940, 593021790883139034940530742, 2301657092451031686369993866, 8938587909568653907369425958, 34733257428574909420620028244, 135039547504477861092309049384, 525298761083875106812869375644, 2044434955451702885538465055648, 7960746028068604806389649761456, 31012747520403304430114739046468, 120871961442804947982331883910548, 471305992602953180159112973065052, 1838509915764924713983905719695208, 7174778363482389430941243269795488, 28010746052275269111003701295025544, 109397901105284663065963080626772709, 427421285705876388468576200688782870, 1670556265154271877814783972839774291, 6531592251258560165319736832154616745, 25546148247820976672573758418678925415, 99948391438524619889003448830324522930, 391170425422710554411899388213967683260, 1531411921681627918654723644279956575070, 5997220252656016599568114009679222560540, 23492879732660759903547886529604541774880, 92054974581536269776798112132779307019110, 360810886949956328160574037104728687493910, 1414588481538225498019554509186410081918210, 5547478061865468582825789313334664125164940, 21760710033190618673331256264861566517464640, 85380685930776506057949552465079389345553100, 335083121816495804325513634015829335000310770, 1315373636637777120797665579902684602111012140, 5164720349429189653534069339907218025696928930, 20283510030009583889803023173897336215483453230, 79677490708326200672964407675113318174062520450, 313056065678144891004061923508475029846054184220, 1230269668298182032984035529737167555169750997720, 4835797311464899664438488946364211209249306752980, 19011806023311065416210522266447470023009490751120, 74759119288493595061238482120440564572696519411760, 294027916914761835601047934178320730449650578632860, 1156630613368748676516639323954238191236287877504940, 4550726362775890758689714652902967105927546572024740, 17907911579394254217754905177886815754307287838553880, 70483206725225504043539019391567643663606387726822560, 277460541353122721063685469530957863426979646844907320, 1092421111771767940115318293241746520292838025303509290, 4301804696225559381602545400193388535286957856783736780, 16942658606728950226364286647782903183819399126525535710, 66739240227665412949130078845505306711774076604133540970, 262934993872915561610432439388195094536133603450338452630, 1036052117468828899418036575972433756017045543114618391700

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

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

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

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

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

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 58
a(6) = 181
a(7) = 578
a(8) = 1888
a(9) = 6297
a(n+6) = 4*(-5+2*n)/(n+7)*a(n)-2*(-8+7*n)/(n+7)*a(1+n)-(13+n)/(n+7)*a(n+2)+(67+21*n)/(n+7)*a(n+3)-(95+21*n)/(n+7)*a(n+4)+2*(23+4*n)/(n+7)*a(n+5), 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) = 58
a \! \left(6\right) = 181
a \! \left(7\right) = 578
a \! \left(8\right) = 1888
a \! \left(9\right) = 6297
a \! \left(n +6\right) = \frac{4 \left(-5+2 n \right) a \! \left(n \right)}{n +7}-\frac{2 \left(-8+7 n \right) a \! \left(1+n \right)}{n +7}-\frac{\left(13+n \right) a \! \left(n +2\right)}{n +7}+\frac{\left(67+21 n \right) a \! \left(n +3\right)}{n +7}-\frac{\left(95+21 n \right) a \! \left(n +4\right)}{n +7}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{n +7}, \quad n \geq 10

Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/21647/
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[30,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[21,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[8,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[8,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]+F[6,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[8,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[8,x]
F[29,x] = F[27,x]+F[6,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[5,x]*F[8,x]
F[32,x] = F[33,x,1]
F[33,x,k[0]] = (F[34,x,k[0]]*k[0]-F[34,x,1])/(-1+k[0])
F[34,x,k[0]] = F[35,x,k[0]]+F[39,x,k[0]]
F[35,x,k[0]] = F[1,x]+F[36,x,k[0]]
F[36,x,k[0]] = F[37,x,k[0]]
F[37,x,k[0]] = F[35,x,k[0]]*F[38,x,k[0]]
F[38,x,k[0]] = k[0]*x
F[39,x,k[0]] = F[40,x,k[0]]
F[40,x,k[0]] = F[33,x,k[0]]*F[35,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_{30}\! \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_{21}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right)+F_{6}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x , 1\right)
F_{33}\! \left(x , y\right) = \frac{y F_{34}\! \left(x , y\right)-F_{34}\! \left(x , 1\right)}{-1+y}
F_{34}\! \left(x , y\right) = F_{35}\! \left(x , y\right)+F_{39}\! \left(x , y\right)
F_{35}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{36}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = F_{37}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = F_{35}\! \left(x , y\right) F_{38}\! \left(x , y\right)
F_{38}\! \left(x , y\right) = y x
F_{39}\! \left(x , y\right) = F_{40}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{33}\! \left(x , y\right) F_{35}\! \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_30(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_21(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_8(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_8(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x) + F_6(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_8(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_8(x))
Eq(F_29(x), F_27(x) + F_6(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_5(x)*F_8(x))
Eq(F_32(x), F_33(x, 1))
Eq(F_33(x, k_0), (-k_0*F_34(x, k_0) + F_34(x, 1))/(1 - k_0))
Eq(F_34(x, k_0), F_35(x, k_0) + F_39(x, k_0))
Eq(F_35(x, k_0), F_1(x) + F_36(x, k_0))
Eq(F_36(x, k_0), F_37(x, k_0))
Eq(F_37(x, k_0), F_35(x, k_0)*F_38(x, k_0))
Eq(F_38(x, k_0), k_0*x)
Eq(F_39(x, k_0), F_40(x, k_0))
Eq(F_40(x, k_0), F_33(x, k_0)*F_35(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, 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, 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, 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, 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