0123_0132_0213_0231_0312_1023_1203

Counting sequence:
1, 1, 2, 6, 17, 51, 161, 524, 1747, 5939, 20510, 71756, 253797, 906032, 3260380, 11814305, 43071858, 157876851, 581469125, 2150809098, 7986558110, 29760593273, 111252691988, 417106577014, 1568001345892, 5909033613693, 22319035449397, 84479358962453, 320388591540279, 1217296607569052, 4632934842881761, 17660740524659481, 67423533626496182, 257765522421078187, 986763627351983024, 3782193467304711541, 14513985106984433567, 55758922469482975924, 214437849801029280658, 825513987449356160023, 3180985207803811975360, 12268542273803490613800, 47358690962575773450482, 182963490224242685857579, 707409672596188464028535, 2737185483156233026613997, 10598636278127450408240805, 41067193432859036628838442, 159230738157710486216366517, 617780184745644364873113698, 2398309628150425961133906215, 9316019980920211718044265105, 36207598589990435913326278126, 140800435569701118748149296254, 547815813735295918358114760723, 2132471410679574774386805624962, 8305048830491819470754517799126, 32359673046946639578470458041584, 126142676789194882249651284959001, 491936807268973865953212227530647, 1919285829745334628703426237664495, 7491125495872872893436255615145065, 29249999867234755622177983180169282, 114253790585576871407774725515944966, 446453412640224311839435834177762169, 1745169129682889344790875406720972647, 6824173058971335359962750503747971077, 26693724604127416931204267473090896117, 104450530657065120360618887000967445394, 408837139851299890173561822717180255163, 1600752937897440870308489116065428046398, 6269440360747035208005885190608100981005, 24561796452424770480447228816921148013115, 96253137880802883574889074439332100396792, 377302560738868622744151417776488151377104, 1479385988411972010056336343347527249278035, 5802124879156217226553508778891303393562472, 22761638145691733709906069018852424118650670, 89315738368536193005073438106335522880508742, 350556299990815537604700171033861459851571637, 1376227519933455664759322814901613896603941221, 5404093412969718237757541547350662503546194247, 21225267228298383821900595874361422280521651283, 83383255285136646999132105940801016140062551992, 327640529363618318553949195184260779739482822925, 1287678069435429336998852151833453190821900135314, 5061809268075814168612391457050646233911835432941, 19901738411507519480657736666249301438218666409781, 78263820005311711929890337871280837111765799296294, 307832138251527466952598598126553314274421528762958, 1211010550358945925224101418949629039642120719949229, 4764980927332388953780053499572464266515638048063932, 18752188429901429580459686538116810446496023026655580, 73810583205909544642827398094438327000934085989226558, 290575899136316634132728027186510143306594677832031369, 1144124426309933779497361611533830270767411354947856371, 4505656989531430472384232188625416556423513501124061953, 17746500707812657013616165284575290046437942414373781803, 69909412103620791602466544029560049784014559056428931828, 275439046412044468894573270091221526138125900425174314291, 1085377870375004309686613720525619498768531521974694909002

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

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

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

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

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

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 51
a(6) = 161
a(n+7) = 2*(5+2*n)/(n+8)*a(n)-(n+4)/(n+8)*a(1+n)+2*(23+6*n)/(n+8)*a(n+2)-(26+7*n)/(n+8)*a(n+3)+(66+13*n)/(n+8)*a(n+4)-(46+7*n)/(n+8)*a(n+5)+(34+5*n)/(n+8)*a(n+6), n >= 7

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) = 51
a \! \left(6\right) = 161
a \! \left(n +7\right) = \frac{2 \left(5+2 n \right) a \! \left(n \right)}{n +8}-\frac{\left(n +4\right) a \! \left(1+n \right)}{n +8}+\frac{2 \left(23+6 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(26+7 n \right) a \! \left(n +3\right)}{n +8}+\frac{\left(66+13 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(46+7 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(34+5 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7

Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/22693/
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[12,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[11,x]
F[10,x] = F[1,x]+F[8,x]
F[11,x] = F[6,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[5,x]*F[8,x]
F[14,x] = F[15,x]+F[4,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[18,x,1]
F[18,x,k[0]] = (F[19,x,k[0]]*k[0]-F[19,x,1])/(-1+k[0])
F[19,x,k[0]] = F[20,x,k[0]]+F[24,x,k[0]]
F[20,x,k[0]] = F[0,x]+F[21,x,k[0]]
F[21,x,k[0]] = F[22,x,k[0]]
F[22,x,k[0]] = F[19,x,k[0]]*F[23,x,k[0]]
F[23,x,k[0]] = k[0]*x
F[24,x,k[0]] = F[25,x,k[0]]
F[25,x,k[0]] = F[18,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_{12}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x , 1\right)
F_{18}\! \left(x , y\right) = \frac{y F_{19}\! \left(x , y\right)-F_{19}\! \left(x , 1\right)}{-1+y}
F_{19}\! \left(x , y\right) = F_{20}\! \left(x , y\right)+F_{24}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{19}\! \left(x , y\right) F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = y x
F_{24}\! \left(x , y\right) = F_{25}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{18}\! \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_12(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_11(x))
Eq(F_10(x), F_1(x) + F_8(x))
Eq(F_11(x), F_6(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_5(x)*F_8(x))
Eq(F_14(x), F_15(x) + F_4(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_18(x, 1))
Eq(F_18(x, k_0), (-k_0*F_19(x, k_0) + F_19(x, 1))/(1 - k_0))
Eq(F_19(x, k_0), F_20(x, k_0) + F_24(x, k_0))
Eq(F_20(x, k_0), F_0(x) + F_21(x, k_0))
Eq(F_21(x, k_0), F_22(x, k_0))
Eq(F_22(x, k_0), F_19(x, k_0)*F_23(x, k_0))
Eq(F_23(x, k_0), k_0*x)
Eq(F_24(x, k_0), F_25(x, k_0))
Eq(F_25(x, k_0), F_18(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 0, 3], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 0, 3], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 0, 3], "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], [0, 0]]}, {"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, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [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, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "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", 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