0231_1032_1320_2031

Counting sequence:
1, 1, 2, 6, 20, 67, 224, 752, 2544, 8683, 29909, 103965, 364595, 1289413, 4596270, 16504417, 59664523, 217018664, 793776629, 2918045776, 10776313029, 39961759773, 148746642987, 555556760411, 2081402323013, 7820111955902, 29457498241200, 111227780271014, 420905101907792, 1596013896575823, 6063265886563582, 23074736376616797, 87958074210716613, 335797370205401655, 1283807042507903579, 4914793712268122118, 18839073690336034183, 72298813384551641332, 277773945335727364772, 1068351765578523221773, 4113157026758341805495, 15850839671231163173881, 61139886170082633043558, 236033296093269485412427, 911969897396700416936872, 3526386057951576891416108, 13646019477491652146695123, 52843865268203152614046581, 204777500289296110110835235, 794065691912738310288581794, 3081097243890547099022489455, 11962394349280918067526107240, 46471231132170073852428786230, 180631636693137436621120712793, 702486227187447882485266892735, 2733426923500539706392646562193, 10641296794621992272787065861700, 41446828188700026924006075500159, 161506622663702025858934193251618, 629628934391121133785513369763828, 2455656978712539844643717891802713, 9581491426726598263700868661358793, 37400287166774462102533454525063279, 146045263067791603818303887555160409, 570512694143369308354344446151968156, 2229478472290650491731841244303604646, 8715583415007036290003825013615606266, 34083183563314717879539025086265807229, 133330609121180149400998640019297159766, 521748611325462541117262203884031753782, 2042350921183485693966264936905857408202, 7997111049272559724563522612112092940101, 31323178387118829534033621213899396765803, 122722686067120985203472160114018435883294, 480957560113957129264606471212725181793534, 1885420471870587409789272213375125221777823, 7393091082842120782933911307995448402516056, 28997276745526909141585599391803305905271051, 113762401155573932855245494152711189506274252, 446424315375279424501930110788855597161646539, 1752272999996601944181526851917286615550015245, 6879516516107804817246391979650525609878688907, 27015547132950689280597038433093487440583797717, 106112584351469026657680811167627508332948164893, 416883735047773831511886437008677627865858331986, 1638157633560080254721850247706516036833676891680, 6438533084992005794429472910249496665383776247986, 25310841035074385279397093714837667632803779400208, 99520510613459518263187632788536678585070284508087, 391383967042186717534442093754960756640222711433086, 1539486870167209156340438184501222710991468010199048, 6056610440997587559412359122117767947456040876215654, 23832094713635026366898278913211304739834783040882925, 93793340386905433053598280389025738611791858190651657, 369196304546051268192708610955197814334735778918808472, 1453505362963596622381610330439472744585369058522157494, 5723323964116281936112790105717025319755023782531792955, 22539844388359637915462732428539028610281445558582676436, 88781592730333131895856942207865677149453947352443413139, 349754226518083129266860798509773476342633271889116516816, 1378064866381210682690321600268556144907757398601485735519

Generating function in Maple syntax:
(-2*(x-1)^5*(x-1/2)*(1-4*x)^(1/2)-4*x^6+3*x^5+7*x^4-20*x^3+18*x^2-7*x+1)/(8*x^7-42*x^6+82*x^5-88*x^4+52*x^3-16*x^2+2*x)

Generating function in latex syntax:
\frac{-2 \left(x -1\right)^{5} \left(x -\frac{1}{2}\right) \sqrt{1-4 x}-4 x^{6}+3 x^{5}+7 x^{4}-20 x^{3}+18 x^{2}-7 x +1}{8 x^{7}-42 x^{6}+82 x^{5}-88 x^{4}+52 x^{3}-16 x^{2}+2 x}

Generating function in sympy syntax:
(-4*x**6 + 3*x**5 + 7*x**4 - 20*x**3 + 18*x**2 - 7*x - 2*sqrt(1 - 4*x)*(x - 1)**5*(x - 1/2) + 1)/(8*x**7 - 42*x**6 + 82*x**5 - 88*x**4 + 52*x**3 - 16*x**2 + 2*x)

Implicit equation for the generating function in Maple syntax:
x*(4*x^4-9*x^3+10*x^2-5*x+1)*(x^2-3*x+1)^2*F(x)^2+(x^2-3*x+1)*(4*x^6-3*x^5-7*x^4+20*x^3-18*x^2+7*x-1)*F(x)+x^8-8*x^7+36*x^6-82*x^5+111*x^4-89*x^3+41*x^2-10*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)^{2}+\left(x^{2}-3 x +1\right) \left(4 x^{6}-3 x^{5}-7 x^{4}+20 x^{3}-18 x^{2}+7 x -1\right) F \! \left(x \right)+x^{8}-8 x^{7}+36 x^{6}-82 x^{5}+111 x^{4}-89 x^{3}+41 x^{2}-10 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 224
a(7) = 752
a(8) = 2544
a(n+9) = 16*(2*n+1)/(n+10)*a(n)-4*(56*n+87)/(n+10)*a(1+n)+2*(325*n+883)/(n+10)*a(n+2)-3*(359*n+1352)/(n+10)*a(n+3)+4*(283*n+1358)/(n+10)*a(n+4)-(777*n+4534)/(n+10)*a(n+5)+2*(173*n+1188)/(n+10)*a(n+6)-2*(48*n+379)/(n+10)*a(n+7)+(134+15*n)/(n+10)*a(n+8), 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) = 20
a \! \left(5\right) = 67
a \! \left(6\right) = 224
a \! \left(7\right) = 752
a \! \left(8\right) = 2544
a \! \left(n +9\right) = \frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +10}-\frac{4 \left(56 n +87\right) a \! \left(1+n \right)}{n +10}+\frac{2 \left(325 n +883\right) a \! \left(n +2\right)}{n +10}-\frac{3 \left(359 n +1352\right) a \! \left(n +3\right)}{n +10}+\frac{4 \left(283 n +1358\right) a \! \left(n +4\right)}{n +10}-\frac{\left(777 n +4534\right) a \! \left(n +5\right)}{n +10}+\frac{2 \left(173 n +1188\right) a \! \left(n +6\right)}{n +10}-\frac{2 \left(48 n +379\right) a \! \left(n +7\right)}{n +10}+\frac{\left(134+15 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21445/
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[14,x] = F[12,x]*F[15,x]*F[9,x]
F[15,x] = F[16,x]+F[36,x]
F[16,x] = F[17,x]*F[30,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[18,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[1,x]+F[23,x]
F[25,x] = F[26,x]+F[27,x]+F[29,x]
F[26,x] = 0
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[19,x]+F[25,x]
F[29,x] = F[12,x]*F[21,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[17,x]*F[30,x]
F[36,x] = F[24,x]^2*F[2,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_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{36}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{30}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right)+F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{30}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24} \left(x \right)^{2} F_{2}\! \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))
Eq(F_14(x), F_12(x)*F_15(x)*F_9(x))
Eq(F_15(x), F_16(x) + F_36(x))
Eq(F_16(x), F_17(x)*F_30(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_18(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_1(x) + F_23(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_29(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_19(x) + F_25(x))
Eq(F_29(x), F_12(x)*F_21(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_17(x)*F_30(x))
Eq(F_36(x), F_2(x)*F_24(x)**2)
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, 2, 3, 1], "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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 2, 3, 1], "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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 2, 3, 1], "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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 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