0231_0312_1203_2031_2301

Counting sequence:
1, 1, 2, 6, 19, 57, 161, 435, 1143, 2958, 7604, 19517, 50154, 129195, 333712, 864218, 2243133, 5833109, 15191491, 39611279, 103381361, 270009866, 705601222, 1844698441, 4824301844, 12619820807, 33018385986, 86401785670, 226119865463, 591823596673, 1549082493189, 4054887016507, 10614504819499, 27786479963830, 72740640110712, 190426850440293, 498522731348158, 1305106983873619, 3416729500804436, 8944944079595394, 23417827860084721, 61307989744855581, 160505041862865767, 420204936820498551, 1100105370552132069, 2880102378742889682, 7540184173490507786, 19740414957356561105, 51680990329835015208, 135302415294660147663, 354225974079168736774, 927374943992892662222, 2427897731999602429419, 6356316000206100964265, 16641045765019073117737, 43566812287651863674243, 114059373083538008450783, 298611270934165142743454, 781774367661363381882556, 2046711687934736927080909, 5358360407912471247682706, 14028368959341924512579771, 36726745317191797683247512, 96151864686390459323508586, 251728844130293561859932069, 659034658481118189401555557, 1725375112866316932635228811, 4517090643224344461085075615, 11825896743019740155781841753, 30960599438260923416584089178, 81055901276615124914617654734, 212207103801288640968563280537, 555565408946659177273660943164, 1454489120677505649417597004375, 3807901948363491288109484921162, 9969216714968235249171567399318, 26099748177651748527926636492383, 68330027780208078471651180441489, 178890335087414623161112581489101, 468340977330920063559857917266699, 1226132596603114112614803876716691, 3210056811873959364477239125615686, 8404037837809838161202284325512832, 22002056699137703479900355501602421, 57602132254767569000040265480565094, 150804340055493596963303407542542307, 394810887892370408776035890351865436, 1033628323582932003137136129922562450, 2706074082779054348180036232234736473, 7084593924599488536492300032419370093, 18547707690709926251475518796298710319, 48558529146911320198291566219027320327, 127127879748786094304113799585884252109, 332825110096971082635479071988827317410, 871347450537175393445181895281001337362, 2281217241504651577385783571654983839393, 5972304273956972298083602735285564336208, 15635695580326651235607892465404937341599, 40934782466943753246225810323335703890446, 107168651820346152178041009829415086586014, 280571172993777790637840161814535380228051

Generating function in Maple syntax:
-(3*x^6-18*x^5+38*x^4-43*x^3+26*x^2-8*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^4

Generating function in latex syntax:
-\frac{3 x^{6}-18 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}

Generating function in sympy syntax:
(-3*x**6 + 18*x**5 - 38*x**4 + 43*x**3 - 26*x**2 + 8*x - 1)/((x - 1)**4*(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)*(x-1)^4*F(x)+3*x^6-18*x^5+38*x^4-43*x^3+26*x^2-8*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+3 x^{6}-18 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0

Explicit closed form in Maple syntax:
-1/6*n^3-11/6*n-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)+2^n

Explicit closed form in latex syntax:
-\frac{n^{3}}{6}-\frac{11 n}{6}-\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}+2^{n}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(6) = 161
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)-1/6*(n-2)*(n^2+2*n+3), 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) = 19
a \! \left(5\right) = 57
a \! \left(6\right) = 161
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{\left(n -2\right) \left(n^{2}+2 n +3\right)}{6}, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21983/
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[20,x]+F[8,x]
F[8,x] = F[13,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[9,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[16,x]+F[17,x]+F[19,x]
F[16,x] = 0
F[17,x] = F[12,x]*F[18,x]
F[18,x] = F[10,x]+F[15,x]
F[19,x] = F[12,x]*F[14,x]
F[20,x] = F[21,x]
F[21,x] = F[9,x]^2*F[22,x]
F[22,x] = F[23,x]+F[31,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[1,x]+F[23,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[23,x]+F[28,x]
F[31,x] = F[16,x]+F[32,x]+F[33,x]
F[32,x] = F[12,x]*F[22,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[10,x]+F[31,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[10,x]+F[37,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[31,x]+F[40,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_{20}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right) F_{9}\! \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_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)+F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{31}\! \left(x \right)+F_{40}\! \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_20(x) + F_8(x))
Eq(F_8(x), F_13(x)*F_9(x))
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))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_9(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_19(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_12(x)*F_18(x))
Eq(F_18(x), F_10(x) + F_15(x))
Eq(F_19(x), F_12(x)*F_14(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_9(x)**2)
Eq(F_22(x), F_23(x) + F_31(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_1(x) + F_23(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_23(x) + F_28(x))
Eq(F_31(x), F_16(x) + F_32(x) + F_33(x))
Eq(F_32(x), F_12(x)*F_22(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_10(x) + F_31(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_10(x) + F_37(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_31(x) + F_40(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, 1, 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": [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]]}], "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, 1, 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": [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]]}], "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, 1, 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": [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]]}], "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], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 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": 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