0132_0213_0231_0321_1320_3021_3120

Counting sequence:
1, 1, 2, 6, 17, 46, 119, 297, 721, 1713, 4001, 9217, 20993, 47361, 105985, 235521, 520193, 1142785, 2498561, 5439489, 11796481, 25493505, 54919169, 117964801, 252706817, 540016641, 1151336449, 2449473537, 5200936961, 11022630913, 23320330241, 49257906177, 103884521473, 218774896641, 460098371585, 966367641601, 2027224563713, 4247722655745, 8890582302721, 18588618457089, 38826504355841, 81020263071745, 168912473817089, 351843720888321, 732274744098817, 1522823604469761, 3164394464739329, 6570681487589377, 13633944184422401, 28270642973376513, 58581979527905281, 121315714962292737, 251075679225905153, 519321332031160321, 1073545561174441985, 2218022816479969281, 4580160821035794433, 9453055617850671105, 19500586386514247681, 40208137473163788289, 82866233143617126401, 170704440275851280385, 351496943717012471809, 723458244140796477441, 1488421662447439446017, 3061006594731178721281, 6292645572144170795009, 12931167595670395682817, 26563311466141754327041, 54547022225959144128513, 111971736527416978309121, 229772644182126174928897, 471351204571426462892033, 966609389462380504678401, 1981623035374174872797185, 4061235175267894883778561, 8320809742816314866532353, 17043020636676549576228865, 34898288308406678129213441, 71439960152851992792793089, 146204466309644215816028161, 299133582490894806416359425, 611867580452454191048163329, 1251238223301141195820892161, 2558087034304555333678268417, 5228604169833271180604211201, 10684486393754092646053183489, 21828364598961744378494713857, 44585184227387523963163770881, 91046621326816952405471526913, 185884434023945381902821621761, 379428622040969054256581574657, 774331494573005361949402202113, 1579920975137966575840007290881, 3222976892279487545699869917185, 6573461608605369259714349629441, 13404414745382097616607717097473, 27328764307264054948673066369025, 55707301767842112370460590080001, 113533956882940795771548480831489, 231346234541651865773148334981121

Generating function in Maple syntax:
-(x^6-3*x^5+x^4+10*x^3-13*x^2+6*x-1)/(x-1)/(2*x-1)^3

Generating function in latex syntax:
-\frac{x^{6}-3 x^{5}+x^{4}+10 x^{3}-13 x^{2}+6 x -1}{\left(x -1\right) \left(2 x -1\right)^{3}}

Generating function in sympy syntax:
(-x**6 + 3*x**5 - x**4 - 10*x**3 + 13*x**2 - 6*x + 1)/((x - 1)*(2*x - 1)**3)

Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)^3*F(x)+x^6-3*x^5+x^4+10*x^3-13*x^2+6*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right)^{3} F \! \left(x \right)+x^{6}-3 x^{5}+x^{4}+10 x^{3}-13 x^{2}+6 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1+1/64*(n^2+17*n-20)*2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 1+\frac{\left(n^{2}+17 n -20\right) 2^{n}}{64} & \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(5) = 46
a(6) = 119
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-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) = 17
a \! \left(5\right) = 46
a \! \left(6\right) = 119
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-1, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22836/
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[29,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[1,x]+F[10,x]+F[28,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[25,x]*F[8,x]
F[12,x] = F[13,x]+F[16,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[8,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[8,x]
F[19,x] = F[1,x]+F[18,x]
F[20,x] = F[21,x]+F[22,x]+F[24,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[8,x]
F[23,x] = F[14,x]+F[20,x]
F[24,x] = F[16,x]*F[8,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[13,x]*F[25,x]*F[8,x]
F[28,x] = F[8,x]*F[9,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[34,x]*F[8,x]
F[31,x] = F[1,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[8,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[19,x]*F[25,x]
F[36,x] = F[17,x]+F[37,x]
F[37,x] = F[21,x]+F[38,x]+F[43,x]
F[38,x] = F[39,x]*F[8,x]
F[39,x] = F[14,x]+F[40,x]
F[40,x] = F[21,x]+F[38,x]+F[41,x]
F[41,x] = F[42,x]*F[8,x]
F[42,x] = F[17,x]+F[40,x]
F[43,x] = F[44,x]*F[8,x]
F[44,x] = F[42,x]+F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]*F[8,x]
F[48,x] = F[17,x]+F[46,x]
F[49,x] = 2*F[21,x]+F[50,x]+F[52,x]
F[50,x] = F[51,x]*F[8,x]
F[51,x] = F[40,x]+F[49,x]
F[52,x] = F[45,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_{29}\! \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_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right)+F_{20}\! \left(x \right)
F_{24}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{34}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{19}\! \left(x \right) F_{25}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{8}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right) F_{8}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right) F_{8}\! \left(x \right)
F_{44}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right) F_{8}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right) F_{8}\! \left(x \right)
F_{51}\! \left(x \right) = F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{45}\! \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_29(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_1(x) + F_10(x) + F_28(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_25(x)*F_8(x))
Eq(F_12(x), F_13(x) + F_16(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_8(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_8(x))
Eq(F_19(x), F_1(x) + F_18(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_24(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_8(x))
Eq(F_23(x), F_14(x) + F_20(x))
Eq(F_24(x), F_16(x)*F_8(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_13(x)*F_25(x)*F_8(x))
Eq(F_28(x), F_8(x)*F_9(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_34(x)*F_8(x))
Eq(F_31(x), F_1(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_8(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_19(x)*F_25(x))
Eq(F_36(x), F_17(x) + F_37(x))
Eq(F_37(x), F_21(x) + F_38(x) + F_43(x))
Eq(F_38(x), F_39(x)*F_8(x))
Eq(F_39(x), F_14(x) + F_40(x))
Eq(F_40(x), F_21(x) + F_38(x) + F_41(x))
Eq(F_41(x), F_42(x)*F_8(x))
Eq(F_42(x), F_17(x) + F_40(x))
Eq(F_43(x), F_44(x)*F_8(x))
Eq(F_44(x), F_42(x) + F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x)*F_8(x))
Eq(F_48(x), F_17(x) + F_46(x))
Eq(F_49(x), 2*F_21(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_51(x)*F_8(x))
Eq(F_51(x), F_40(x) + F_49(x))
Eq(F_52(x), F_45(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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": []}, 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