0231_1032_1203_1302

Counting sequence:
1, 1, 2, 6, 20, 66, 214, 688, 2206, 7070, 22660, 72634, 232830, 746352, 2392486, 7669286, 24584436, 78807122, 252621702, 809796400, 2595858574, 8321204878, 26674199972, 85506000010, 274095419758, 878631898608, 2816515554774, 9028536162646, 28941599524244, 92774306701794, 297394481489846, 953318658633648, 3055929149546494, 9795993063256318, 31401735903990724, 100660444675345370, 322674044301817054, 1034354052397630576, 3315693730576814982, 10628686463307929094, 34070992412092462900, 109216931739614070770, 350102457666654829926, 1122277735804345505840, 3597539202313600002798, 11532161691604200792814, 36967144985043185967844, 118500749893240924300714, 379862381337319122899918, 1217675237374081888265968, 3903342517082139278233270, 12512435449141399664904246, 40109480575628350929952212, 128573724802480514108364418, 412152001779569712691773590, 1321181857582998368420125872, 4235140174667925909691671902, 13576035877376919820284572766, 43518925594541111207261664004, 139502937529738820933082998266, 447186351996414005495721634366, 1433486899652062725776073839984, 4595141784404370874142649752422, 14730046032443071312263618419686, 47218185269993011345423839191412, 151361171260481041037844466028946, 485198743542226411020309929483590, 1555338259967736813218333741980976, 4985744780085023353100877067635534, 15982151054819864717057288862245198, 51231894853375107366453639805701156, 164227396003476577225270685263331530, 526442320263036593429407103125730350, 1687547408704347356035955278008135664, 5409550385694386935075009409213891990, 17340689348593649556840988512515431702, 55586783677924928662823697810112457236, 178187294492246644156531607099809832226, 571191744110133318370256104878323687286, 1830994793816639797406854065933399439792, 5869380938281252335068604023917970716862, 18814708111130316366976066077569352254910, 60311853162949374679027518484154564015684, 193333832789959668166579545973459237161498, 619745024253673983547609364788491646717342, 1986635704390449739288100076064861829518960, 6368298683336369484273553010698704569219398, 20414023582963395180541501980193456626333638, 65438569949024188469945039130335798465208244, 209767879397722654023028893833119614218583346, 672425501677297531547695203174021308341793574, 2155506632398525833457053915559612436039717424, 6909626167842437062543518730912391478680365486, 22149286419131694604708997058340755574490979758, 71001075450355516919568569837555487265851210084, 227598876989225470615053944703041713222540654058, 729584002470165513057125722885683177549708718734, 2338732175227671338658525280857345844254163855600, 7496968366804099984352962122899113831313020863094, 24032052617307462603383636540036880956984792201206, 77036413219819298481081548963319991342749470386772

Generating function in Maple syntax:
(x-1)*(x^2-3*x+1)/(4*x^3-7*x^2+5*x-1)

Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{2}-3 x +1\right)}{4 x^{3}-7 x^{2}+5 x -1}

Generating function in sympy syntax:
(x - 1)*(x**2 - 3*x + 1)/(4*x**3 - 7*x**2 + 5*x - 1)

Implicit equation for the generating function in Maple syntax:
(4*x^3-7*x^2+5*x-1)*F(x)-(x-1)*(x^2-3*x+1) = 0

Implicit equation for the generating function in latex syntax:
\left(4 x^{3}-7 x^{2}+5 x -1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{2}-3 x +1\right) = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/171336*(((1947*I+110*59^(1/2))*3^(1/2)-330*I*59^(1/2)-1947)*(71+6*59^(1/2)*3^(1/2))^(1/3)+42834+((-177*I+32*59^(1/2))*3^(1/2)+96*I*59^(1/2)-177)*(71+6*59^(1/2)*3^(1/2))^(2/3))*(1/2904*((71*I-6*59^(1/2))*3^(1/2)-18*I*59^(1/2)+71)*(71+6*59^(1/2)*3^(1/2))^(2/3)-1/24*I*3^(1/2)*(71+6*59^(1/2)*3^(1/2))^(1/3)+1/24*(71+6*59^(1/2)*3^(1/2))^(1/3)+7/12)^(-n)+1/171336*(((177*I+32*59^(1/2))*3^(1/2)-96*I*59^(1/2)-177)*(71+6*59^(1/2)*3^(1/2))^(2/3)+42834+((-1947*I+110*59^(1/2))*3^(1/2)+330*I*59^(1/2)-1947)*(71+6*59^(1/2)*3^(1/2))^(1/3))*(1/2904*((-71*I-6*59^(1/2))*3^(1/2)+18*I*59^(1/2)+71)*(71+6*59^(1/2)*3^(1/2))^(2/3)+1/24*I*3^(1/2)*(71+6*59^(1/2)*3^(1/2))^(1/3)+1/24*(71+6*59^(1/2)*3^(1/2))^(1/3)+7/12)^(-n)-5/3894*(-1/12*(71+6*59^(1/2)*3^(1/2))^(1/3)+7/12-71/1452*(71+6*59^(1/2)*3^(1/2))^(2/3)+1/242*(71+6*59^(1/2)*3^(1/2))^(2/3)*59^(1/2)*3^(1/2))^(-n)*((16/55*59^(1/2)*3^(1/2)-177/110)*(71+6*59^(1/2)*3^(1/2))^(2/3)+(71+6*59^(1/2)*3^(1/2))^(1/3)*59^(1/2)*3^(1/2)-177/10*(71+6*59^(1/2)*3^(1/2))^(1/3)-1947/10))

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(1947 \,\mathrm{I}+110 \sqrt{59}\right) \sqrt{3}-330 \,\mathrm{I} \sqrt{59}-1947\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+42834+\left(\left(-177 \,\mathrm{I}+32 \sqrt{59}\right) \sqrt{3}+96 \,\mathrm{I} \sqrt{59}-177\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(71 \,\mathrm{I}-6 \sqrt{59}\right) \sqrt{3}-18 \,\mathrm{I} \sqrt{59}+71\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2904}-\frac{\mathrm{I} \sqrt{3}\, \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{7}{12}\right)^{-n}}{171336}\\+\\\frac{\left(\left(\left(177 \,\mathrm{I}+32 \sqrt{59}\right) \sqrt{3}-96 \,\mathrm{I} \sqrt{59}-177\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+42834+\left(\left(-1947 \,\mathrm{I}+110 \sqrt{59}\right) \sqrt{3}+330 \,\mathrm{I} \sqrt{59}-1947\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-71 \,\mathrm{I}-6 \sqrt{59}\right) \sqrt{3}+18 \,\mathrm{I} \sqrt{59}+71\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2904}+\frac{\mathrm{I} \sqrt{3}\, \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{7}{12}\right)^{-n}}{171336}\\-\\\frac{5 \left(-\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{7}{12}-\frac{71 \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{1452}+\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{59}\, \sqrt{3}}{242}\right)^{-n} \left(\left(\frac{16 \sqrt{59}\, \sqrt{3}}{55}-\frac{177}{110}\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{59}\, \sqrt{3}-\frac{177 \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{10}-\frac{1947}{10}\right)}{3894} & \mathit{\text{otherwise}} \end{array}\right.

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

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(n +3\right) = 4 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 4

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21434/
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[7,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[7,x]*F[8,x]
F[7,x] = x
F[8,x] = F[22,x]+F[9,x]
F[9,x] = F[10,x]+F[4,x]
F[10,x] = F[11,x]
F[11,x] = F[0,x]*F[12,x]
F[12,x] = F[13,x]+F[18,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[7,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[13,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[7,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[0,x]*F[13,x]*F[21,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_{7}\! \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_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = x
F_{8}\! \left(x \right) = F_{22}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{4}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{0}\! \left(x \right) F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{0}\! \left(x \right) F_{13}\! \left(x \right) F_{21}\! \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_7(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_7(x)*F_8(x))
Eq(F_7(x), x)
Eq(F_8(x), F_22(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_4(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_0(x)*F_12(x))
Eq(F_12(x), F_13(x) + F_18(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_7(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_13(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_7(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_0(x)*F_13(x)*F_21(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": [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": [1, 3, 0, 2], "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": [1, 3, 0, 2], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "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": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 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