0231_0321_1302_1320_2031_2301_3021

Counting sequence:
1, 1, 2, 6, 17, 47, 128, 345, 923, 2456, 6509, 17199, 45344, 119345, 313715, 823848, 2161925, 5670119, 14864816, 38957097, 102072011, 267390008, 700360157, 1834214751, 4803332672, 12577880417, 32934502883, 86234016840, 225784324853, 591152512151, 1547740320464, 4052202666969, 10609136115899, 27775742551640, 72719165280845, 190383900774543, 498436832010080, 1304935185190289, 3416385903429971, 8944256884837992, 23416453470560741, 61305240965797703, 160499544304739312, 420193941704234121, 1100083380319590827, 2880058398277793912, 7540096212560302013, 19740239035496134335, 51680638486114145408, 135301711607218390721, 354224566704285204419, 927372129243125577864, 2427892102500068239829, 6356304741207032562935, 16641023247020936291600, 43566767251655589997113, 114059283011545461070235, 298611090790180047954584, 781774007373393192275501, 2046710967358796547835887, 5358358966760590489160096, 14028366077038162995500273, 36726739552584274649052467, 96151853157175413255080616, 251728821071863469723036357, 659034612364258005127722407, 1725375020632596564087518768, 4517090458756903723989609705, 11825896374084858681590861963, 30960598700391160468202079416, 81055899800875599017853582749, 212207100849809589175035081759, 555565403043701073686604488384, 1454489108871589442243484035105, 3807901924751658873761258920355, 9969216667744570420475115332808, 26099748083204418870533732291765, 68330027591313419156865371969879, 178890334709625304531540964472656, 468340976575341426300714683157657, 1226132595091956838096517408419451, 3210056808851644815440666188938968, 8404037831765209063129138452073997, 22002056687048445283754063754636111, 57602132230589052607747681986540512, 150804340007136564178718240554397777, 394810887795656343206865556375477523, 1033628323389503871998795461969684200, 2706074082392198085903354896328873893, 7084593923825776011938937360607535111, 18547707689162501202368793452674926704, 48558529143816470100078115531779635529, 127127879742596394107686898211388760939, 332825110084591682242625269239836209400, 871347450512416592659474289783018991485, 2281217241455133975814368360659019013503, 5972304273857937094940772313293634545920, 15635695580128580829322231621421077618049, 40934782466547612433654488635367984295811, 107168651819553870552898366453479647244552, 280571172992193227387554875062664501388181

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

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

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

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

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

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-1/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+1/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)-1/4*2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-\frac{2^{n}}{4} & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2), n >= 5

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(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 5

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/23235/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[14,x]+F[8,x]
F[8,x] = F[17,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[9,x]
F[13,x] = x
F[14,x] = F[11,x]*F[15,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[13,x]*F[15,x]
F[17,x] = F[18,x]*F[2,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[13,x]*F[18,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_{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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{17}\! \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_{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_{9}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_14(x) + F_8(x))
Eq(F_8(x), F_17(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(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_9(x))
Eq(F_13(x), x)
Eq(F_14(x), F_11(x)*F_15(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_13(x)*F_15(x))
Eq(F_17(x), F_18(x)*F_2(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_13(x)*F_18(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": 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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]]}, {"patt": [2, 3, 0, 1], "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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]]}, {"patt": [2, 3, 0, 1], "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": "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [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, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [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, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, 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