0231_1032_1203_3021

Counting sequence:
1, 1, 2, 6, 20, 61, 172, 465, 1233, 3243, 8503, 22270, 58307, 152646, 399620, 1046196, 2738942, 7170595, 18772798, 49147743, 128670363, 336863265, 881919337, 2308894636, 6044764445, 15825398556, 41431431062, 108468894450, 283975252088, 743456861593, 1946395332448, 5095729135485, 13340792073717, 34926647085351, 91439149181995, 239390800460266, 626733252198407, 1640808956134530, 4295693616204728, 11246271892479168, 29443122061232258, 77083094291217055, 201806160812418322, 528335388146037291, 1383200003625692895, 3621264622731040701, 9480593864567428477, 24820516970971243960, 64980957048346302593, 170122354174067662968, 445386105473856685418, 1166035962247502392350, 3052721781268650490652, 7992129381558449078581, 20923666363406696744020, 54778869708661641152361, 143412942762578226711897, 375459958579073038982115, 982966932974640890233183, 2573440840344849631716118, 6737355588059908004913803, 17638625923834874383023870, 46178522183444715144156332, 120896940626499271049443596, 316512299696053098004172870, 828639958461660022963073371, 2169407575688926970885045542, 5679582768605120889692061495, 14869340730126435698191137123, 38928439421774186204881347993, 101915977535196122916452904913, 266819493183814182544477364740, 698542502016246424716979187237, 1828808012864925091606460194836, 4787881536578528850102401395070, 12534836596870661458700743988106, 32816628254033455525999830566912, 85915048165229705119298747710225, 224928516241655659831896412561288, 588870500559737274376390489971093, 1541682985437556163297275057349373, 4036178455752931215515434682074335, 10566852381821237483249028988870867, 27664378689710781234231652284535426, 72426283687311106219445927864732495, 189614472372222537424106131309659066, 496417133429356506052872466064241632, 1299636927915846980734511266883062680, 3402493650318184436150661334584943178, 8907844023038706327717472736871763543, 23321038418797934547001756876030344058, 61055271233355097313287797891219265155, 159844775281267357392861636797627447847, 418479054610446974865297112501663074741, 1095592388550073567203029700707361772645, 2868298111039773726743791989620422239376, 7509301944569247613028346268153904941577, 19659607722667969112341246814841292581360, 51469521223434659723995394176369972798418, 134748955947636010059644935714268625809718, 352777346619473370454939412966435904626468

Generating function in Maple syntax:
(x^6+2*x^5-3*x^4+6*x^3-9*x^2+5*x-1)/(x^2-3*x+1)/(x-1)^3

Generating function in latex syntax:
\frac{x^{6}+2 x^{5}-3 x^{4}+6 x^{3}-9 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(x**6 + 2*x**5 - 3*x**4 + 6*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**3*(x**2 - 3*x + 1))

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

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

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/2*(9*5^(1/2)-19)*(3/2-1/2*5^(1/2))^(-n)+1/2*(-9*5^(1/2)-19)*(3/2+1/2*5^(1/2))^(-n)+1/2*n^2-13/2*n+12)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(9 \sqrt{5}-19\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{2}+\frac{\left(-9 \sqrt{5}-19\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{2}+\frac{n^{2}}{2}-\frac{13 n}{2}+12 & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 61
a(6) = 172
a(n+2) = -a(n)+3*a(n+1)-1/2*(-1+n)*(n-10), 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) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 172
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{\left(-1+n \right) \left(n -10\right)}{2}, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15859/
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[13,x]*F[4,x]
F[4,x] = F[26,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]+F[16,x]+F[24,x]
F[15,x] = 0
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[13,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[15,x]+F[16,x]+F[22,x]
F[22,x] = F[13,x]*F[23,x]
F[23,x] = F[10,x]+F[21,x]
F[24,x] = F[13,x]*F[25,x]
F[25,x] = F[9,x]
F[26,x] = F[2,x]+F[27,x]
F[27,x] = F[15,x]+F[28,x]+F[48,x]
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[30,x]+F[35,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[13,x]*F[33,x]
F[33,x] = F[10,x]+F[34,x]
F[34,x] = F[32,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[15,x]+F[28,x]+F[37,x]
F[37,x] = F[13,x]*F[38,x]
F[38,x] = F[2,x]+F[36,x]
F[39,x] = F[40,x]
F[40,x] = F[13,x]*F[41,x]
F[41,x] = F[42,x]+F[47,x]
F[42,x] = F[43,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[13,x]
F[46,x] = F[43,x]
F[47,x] = F[40,x]
F[48,x] = F[13,x]*F[49,x]
F[49,x] = F[26,x]+F[50,x]
F[50,x] = F[42,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[13,x]*F[53,x]
F[53,x] = F[50,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{26}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{24}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{9}\! \left(x \right)
F_{26}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{48}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right)
F_{46}\! \left(x \right) = F_{43}\! \left(x \right)
F_{47}\! \left(x \right) = F_{40}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{26}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{42}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{50}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_26(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_16(x) + F_24(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_13(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_15(x) + F_16(x) + F_22(x))
Eq(F_22(x), F_13(x)*F_23(x))
Eq(F_23(x), F_10(x) + F_21(x))
Eq(F_24(x), F_13(x)*F_25(x))
Eq(F_25(x), F_9(x))
Eq(F_26(x), F_2(x) + F_27(x))
Eq(F_27(x), F_15(x) + F_28(x) + F_48(x))
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_35(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_13(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_34(x))
Eq(F_34(x), F_32(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_15(x) + F_28(x) + F_37(x))
Eq(F_37(x), F_13(x)*F_38(x))
Eq(F_38(x), F_2(x) + F_36(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_13(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_47(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_13(x))
Eq(F_46(x), F_43(x))
Eq(F_47(x), F_40(x))
Eq(F_48(x), F_13(x)*F_49(x))
Eq(F_49(x), F_26(x) + F_50(x))
Eq(F_50(x), F_42(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_13(x)*F_53(x))
Eq(F_53(x), F_50(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "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, 2, 0, 3], "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, 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, 2, 0, 3], "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": 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