0132_0213_1320_2301_3120

Counting sequence:
1, 1, 2, 6, 19, 51, 115, 232, 439, 805, 1463, 2674, 4955, 9335, 17867, 34652, 67887, 133961, 265647, 528486, 1053555, 2103003, 4201123, 8396496, 16786279, 33564781, 67120615, 134231002, 268450379, 536887615, 1073760443, 2147504324, 4294990175, 8589959825, 17179896927, 34359768782, 68719509987, 137438989731, 274877946387, 549755856696, 1099511674135, 2199023305653, 4398046565143, 8796093080386, 17592186106939, 35184372155911, 70368744249515, 140737488432172, 281474976792719, 562949953508825, 1125899906935823, 2251799813784374, 4503599627475795, 9007199254852715, 18014398509600387, 36028797019089312, 72057594038060487, 144115188075995901, 288230376151859527, 576460752303579306, 1152921504607011115, 2305843009213866703, 4611686018427569563, 9223372036854966676, 18446744073709751999, 36893488147419313441, 73786976294838426815, 147573952589676643742, 295147905179353067459, 590295810358705904435, 1180591620717411567603, 2361183241434822882824, 4722366482869645501815, 9444732965739290728005, 18889465931478581168247, 37778931862957162036242, 75557863725914323759387, 151115727451828647192471, 302231454903657294045067, 604462909807314587736316, 1208925819614629175104495, 2417851639229258349826153, 4835703278458516699254383, 9671406556917033398095366, 19342813113834066795761459, 38685626227668133591077371, 77371252455336267181692515, 154742504910672534362905712, 309485009821345068725314599, 618970019642690137450114445, 1237940039285380274899695783, 2475880078570760549798839674, 4951760157141521099597108235, 9903520314283042199193625695, 19807040628566084398386640507, 39614081257132168796772649572, 79228162514264337593544646687, 158456325028528675187088619441, 316912650057057350374176543007, 633825300114114700748352367726, 1267650600228229401496703994275

Generating function in Maple syntax:
-(2*x^7-5*x^6-4*x^5+4*x^4-8*x^3+10*x^2-5*x+1)/(2*x-1)/(x-1)^4

Generating function in latex syntax:
-\frac{2 x^{7}-5 x^{6}-4 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right)^{4}}

Generating function in sympy syntax:
(-2*x**7 + 5*x**6 + 4*x**5 - 4*x**4 + 8*x**3 - 10*x**2 + 5*x - 1)/((x - 1)**4*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^4*F(x)+2*x^7-5*x^6-4*x^5+4*x^4-8*x^3+10*x^2-5*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+2 x^{7}-5 x^{6}-4 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,-1+2^n+5/6*n^3-9/2*n^2+17/3*n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ -1+2^{n}+\frac{5 n^{3}}{6}-\frac{9 n^{2}}{2}+\frac{17 n}{3} & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 115
a(7) = 232
a(n+1) = 2*a(n)-1/6*(-2+n)*(5*n^2-32*n+9), n >= 8

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) = 51
a \! \left(6\right) = 115
a \! \left(7\right) = 232
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(-2+n \right) \left(5 n^{2}-32 n +9\right)}{6}, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21639/
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[15,x]*F[4,x]
F[4,x] = F[25,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[15,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[15,x]*F[22,x]
F[11,x] = F[12,x]+F[19,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]*F[16,x]
F[15,x] = x
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[15,x]*F[16,x]
F[19,x] = F[20,x]
F[20,x] = F[15,x]*F[16,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[15,x]*F[22,x]
F[25,x] = F[26,x]
F[26,x] = F[15,x]*F[27,x]
F[27,x] = F[28,x]+F[42,x]
F[28,x] = F[29,x]+F[40,x]
F[29,x] = F[30,x]+F[37,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[12,x]*F[16,x]
F[32,x] = F[23,x]*F[33,x]
F[33,x] = F[17,x]+F[34,x]
F[34,x] = F[1,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[15,x]*F[16,x]*F[22,x]
F[37,x] = F[22,x]*F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[15,x]*F[35,x]
F[40,x] = F[41,x]
F[41,x] = F[15,x]*F[22,x]*F[23,x]*F[35,x]
F[42,x] = F[43,x]
F[43,x] = F[15,x]*F[44,x]
F[44,x] = F[45,x]+F[53,x]
F[45,x] = F[16,x]*F[22,x]*F[46,x]
F[46,x] = F[47,x]+F[51,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[22,x]^2
F[49,x] = F[50,x]
F[50,x] = F[15,x]*F[17,x]*F[21,x]*F[22,x]
F[51,x] = F[23,x]*F[52,x]
F[52,x] = F[22,x]+F[23,x]
F[53,x] = F[22,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[22,x]*F[35,x]
F[56,x] = F[17,x]*F[23,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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{25}\! \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_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{22}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{19}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{42}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{22}\! \left(x \right)
F_{37}\! \left(x \right) = F_{22}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right) F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right) F_{22}\! \left(x \right) F_{23}\! \left(x \right) F_{35}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{45}\! \left(x \right) = F_{16}\! \left(x \right) F_{22}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{51}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{22} \left(x \right)^{2}
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{21}\! \left(x \right) F_{22}\! \left(x \right)
F_{51}\! \left(x \right) = F_{23}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{22}\! \left(x \right) F_{35}\! \left(x \right)
F_{56}\! \left(x \right) = F_{17}\! \left(x \right) F_{23}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_25(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_15(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_15(x)*F_22(x))
Eq(F_11(x), F_12(x) + F_19(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x)*F_16(x))
Eq(F_15(x), x)
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_15(x)*F_16(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_15(x)*F_16(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_15(x)*F_22(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_15(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_42(x))
Eq(F_28(x), F_29(x) + F_40(x))
Eq(F_29(x), F_30(x) + F_37(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_12(x)*F_16(x))
Eq(F_32(x), F_23(x)*F_33(x))
Eq(F_33(x), F_17(x) + F_34(x))
Eq(F_34(x), F_1(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_15(x)*F_16(x)*F_22(x))
Eq(F_37(x), F_22(x)*F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_15(x)*F_35(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_15(x)*F_22(x)*F_23(x)*F_35(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_15(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_53(x))
Eq(F_45(x), F_16(x)*F_22(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_51(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_22(x)**2)
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_15(x)*F_17(x)*F_21(x)*F_22(x))
Eq(F_51(x), F_23(x)*F_52(x))
Eq(F_52(x), F_22(x) + F_23(x))
Eq(F_53(x), F_22(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_22(x)*F_35(x))
Eq(F_56(x), F_17(x)*F_23(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": [1, 3, 2, 0], "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, 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": [1, 3, 2, 0], "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, 1, 2, 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