0213_0231_1203_1302_2031_3120

Counting sequence:
1, 1, 2, 6, 18, 50, 131, 330, 808, 1934, 4541, 10486, 23866, 53642, 119271, 262738, 574156, 1246038, 2687985, 5768350, 12322142, 26215986, 55576347, 117442586, 247466288, 520096350, 1090522021, 2281704710, 4764733058, 9932115994, 20669534671, 42949677986, 89120576916, 184683599782, 382252095961, 790273989678, 1632087580326, 3367254368578, 6940667159555, 14293651171050, 29411936053752, 60473139539246, 124244813951117, 255086697657366, 523367534835658, 1073123348724650, 2199023255568311, 4503599627387890, 9218305487292508, 18858823439633654, 38562071809380801, 78812993479005886, 161003686678518766, 328762772798071122, 671036344478230251, 1369094286720658618, 2792231768969736896, 5692549928996337918, 11601272640106430325, 23634890844440397350, 48134472817335897362, 97998327891582030906, 199455420296984566687, 405828369621610177346, 825491797298502478628, 1678653710707569242950, 3412647653636267097001, 6935975771714791457870, 14093312472314097487158, 28629346802397224162914, 58144137320332506750931, 118059162071741130402186, 239660099005634494657416, 486403747735573457075662, 986974594919755849730141, 2002283388736729570677558, 4061235175267894883851866, 8235807146124661252762058, 16698287883427065475708295, 33849922949209616891855250, 68606540263130205664660972, 139026469255682355091298966, 281679715970208597706631057, 570612986858104970461410526, 1155733083551585491019203198, 2340480386773922082231259250, 4738989212889346364848316091, 9594035304461697130468322650, 19420184366289403062480125008, 39304596247310823728047311774, 79537647524085682662268853061, 160932205107099435736886274886, 325578230332055012298469800866, 658584100899822306246334221402, 1332023482271069175791457803631, 2693757525484987478180494454498, 5446936172855673209556146733236, 11012714589482742925502609248998, 22263113666508278863785850201465, 45001596308102143753132963952750, 90953930566375459557388455152582

Generating function in Maple syntax:
(x^6-5*x^5+19*x^4-28*x^3+20*x^2-7*x+1)/(2*x-1)^2/(x-1)^4

Generating function in latex syntax:
\frac{x^{6}-5 x^{5}+19 x^{4}-28 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}

Generating function in sympy syntax:
(x**6 - 5*x**5 + 19*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)

Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^4*F(x)-x^6+5*x^5-19*x^4+28*x^3-20*x^2+7*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)-x^{6}+5 x^{5}-19 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/12*(9*n-39)*2^n+1/6*n^3+11/6*n+4)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(9 n -39\right) 2^{n}}{12}+\frac{n^{3}}{6}+\frac{11 n}{6}+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) = 18
a(5) = 50
a(6) = 131
a(n+2) = 1/6*n^3-n^2-4*a(n)+4*a(n+1)+11/6*n+1, 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) = 18
a \! \left(5\right) = 50
a \! \left(6\right) = 131
a \! \left(n +2\right) = \frac{n^{3}}{6}-n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{11 n}{6}+1, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22581/
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[8,x]
F[4,x] = F[31,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[18,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[8,x]
F[22,x] = F[11,x]+F[19,x]
F[23,x] = F[24,x]*F[8,x]
F[24,x] = F[25,x]+F[27,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[23,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[15,x]+F[28,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[36,x]*F[5,x]*F[8,x]
F[33,x] = F[1,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[17,x]*F[33,x]*F[8,x]
F[36,x] = F[17,x]+F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]*F[8,x]
F[40,x] = F[1,x]+F[38,x]
F[41,x] = F[20,x]+F[42,x]+F[44,x]
F[42,x] = F[43,x]*F[8,x]
F[43,x] = F[15,x]+F[41,x]
F[44,x] = F[37,x]*F[8,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{31}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \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_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{36}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right) F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{20}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right) F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \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_8(x))
Eq(F_4(x), F_31(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_18(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_8(x))
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_8(x))
Eq(F_22(x), F_11(x) + F_19(x))
Eq(F_23(x), F_24(x)*F_8(x))
Eq(F_24(x), F_25(x) + F_27(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_23(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_15(x) + F_28(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_36(x)*F_5(x)*F_8(x))
Eq(F_33(x), F_1(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_17(x)*F_33(x)*F_8(x))
Eq(F_36(x), F_17(x) + F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x)*F_8(x))
Eq(F_40(x), F_1(x) + F_38(x))
Eq(F_41(x), F_20(x) + F_42(x) + F_44(x))
Eq(F_42(x), F_43(x)*F_8(x))
Eq(F_43(x), F_15(x) + F_41(x))
Eq(F_44(x), F_37(x)*F_8(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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"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, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], 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