0213_1032_2301_3120

Counting sequence:
1, 1, 2, 6, 20, 56, 134, 290, 592, 1172, 2290, 4462, 8716, 17104, 33726, 66778, 132648, 264108, 526698, 1051494, 2100644, 4198440, 8393462, 16782866, 33560960, 67116356, 134226274, 268445150, 536881852, 1073754112, 2147497390, 4294982602, 8589951576, 17179887964, 34359759066, 68719499478, 137438978388, 274877934168, 549755843558, 1099511660034, 2199023290544, 4398046548980, 8796093063122, 17592186088526, 35184372136300, 70368744228656, 140737488410014, 281474976769210, 562949953483912, 1125899906909452, 2251799813756490, 4503599627446342, 9007199254821636, 18014398509567624, 36028797019054806, 72057594038024178, 144115188075957728, 288230376151819428, 576460752303537218, 1152921504606966974, 2305843009213820444, 4611686018427521120, 9223372036854915982, 18446744073709698986, 36893488147419258040, 73786976294838368956, 147573952589676583354, 295147905179353004470, 590295810358705838772, 1180591620717411499192, 2361183241434822811590, 4722366482869645427682, 9444732965739290650896, 18889465931478581088084, 37778931862957161952946, 75557863725914323672878, 151115727451828647102668, 302231454903657293951888, 604462909807314587639678, 1208925819614629175004314, 2417851639229258349722344, 4835703278458516699146860, 9671406556917033397984042, 19342813113834066795646246, 38685626227668133590958180, 77371252455336267181569256, 154742504910672534362778294, 309485009821345068725182930, 618970019642690137449978432, 1237940039285380274899555332, 2475880078570760549798694690, 4951760157141521099596958622, 9903520314283042199193471356, 19807040628566084398386481344, 39614081257132168796772485486, 79228162514264337593544477578, 158456325028528675187088445208, 316912650057057350374176363548, 633825300114114700748352182938, 1267650600228229401496703804054, 2535301200456458802993407028244

Generating function in Maple syntax:
(2*x^6+5*x^5-5*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^{6}+5 x^{5}-5 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**6 + 5*x**5 - 5*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^6+5*x^5-5*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^{6}+5 x^{5}-5 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,-8+2^(n+1)+2/3*n^3-5*n^2+25/3*n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -8+2^{n +1}+\frac{2 n^{3}}{3}-5 n^{2}+\frac{25 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) = 20
a(5) = 56
a(6) = 134
a(n+1) = -2/3*n^3+7*n^2+2*a(n)-49/3*n+12, 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) = 20
a \! \left(5\right) = 56
a \! \left(6\right) = 134
a \! \left(n +1\right) = -\frac{2 n^{3}}{3}+7 n^{2}+2 a \! \left(n \right)-\frac{49 n}{3}+12, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21384/
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[21,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[16,x]
F[11,x] = F[12,x]+F[18,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[15,x]*F[16,x]
F[18,x] = F[19,x]
F[19,x] = F[15,x]*F[16,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[15,x]*F[23,x]
F[23,x] = F[24,x]+F[48,x]
F[24,x] = F[25,x]+F[29,x]
F[25,x] = F[16,x]*F[26,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[15,x]*F[26,x]
F[29,x] = F[30,x]
F[30,x] = F[15,x]*F[16,x]*F[31,x]
F[31,x] = F[32,x]+F[45,x]
F[32,x] = F[33,x]+F[44,x]
F[33,x] = F[26,x]*F[34,x]
F[34,x] = F[35,x]+F[43,x]
F[35,x] = F[26,x]+F[36,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[17,x]
F[38,x] = F[39,x]+F[40,x]+F[42,x]
F[39,x] = 0
F[40,x] = F[15,x]*F[41,x]
F[41,x] = F[27,x]+F[38,x]
F[42,x] = F[15,x]*F[36,x]
F[43,x] = F[26,x]*F[37,x]
F[44,x] = F[16,x]*F[26,x]*F[27,x]
F[45,x] = F[26,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[15,x]*F[16,x]*F[20,x]*F[27,x]
F[48,x] = F[49,x]
F[49,x] = F[15,x]*F[16,x]*F[50,x]
F[50,x] = F[51,x]+F[57,x]
F[51,x] = F[26,x]*F[52,x]
F[52,x] = F[53,x]+F[55,x]
F[53,x] = F[16,x]*F[54,x]
F[54,x] = F[26,x]+F[27,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[15,x]*F[16,x]*F[27,x]
F[57,x] = F[26,x]*F[58,x]
F[58,x] = F[37,x]+F[59,x]
F[59,x] = F[39,x]+F[60,x]+F[61,x]
F[60,x] = F[15,x]*F[58,x]
F[61,x] = F[15,x]*F[62,x]
F[62,x] = F[27,x]+F[59,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_{21}\! \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_{16}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \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_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{48}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{45}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{44}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{43}\! \left(x \right)
F_{35}\! \left(x \right) = F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{17}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = 0
F_{40}\! \left(x \right) = F_{15}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{27}\! \left(x \right)+F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right) F_{36}\! \left(x \right)
F_{43}\! \left(x \right) = F_{26}\! \left(x \right) F_{37}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right) F_{26}\! \left(x \right) F_{27}\! \left(x \right)
F_{45}\! \left(x \right) = F_{26}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{20}\! \left(x \right) F_{27}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{57}\! \left(x \right)
F_{51}\! \left(x \right) = F_{26}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{16}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{57}\! \left(x \right) = F_{26}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{37}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{39}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right) F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{27}\! \left(x \right)+F_{59}\! \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_21(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_16(x))
Eq(F_11(x), F_12(x) + F_18(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_15(x)*F_16(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_15(x)*F_16(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_15(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_48(x))
Eq(F_24(x), F_25(x) + F_29(x))
Eq(F_25(x), F_16(x)*F_26(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_15(x)*F_26(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_15(x)*F_16(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_45(x))
Eq(F_32(x), F_33(x) + F_44(x))
Eq(F_33(x), F_26(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_43(x))
Eq(F_35(x), F_26(x) + F_36(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_17(x))
Eq(F_38(x), F_39(x) + F_40(x) + F_42(x))
Eq(F_39(x), 0)
Eq(F_40(x), F_15(x)*F_41(x))
Eq(F_41(x), F_27(x) + F_38(x))
Eq(F_42(x), F_15(x)*F_36(x))
Eq(F_43(x), F_26(x)*F_37(x))
Eq(F_44(x), F_16(x)*F_26(x)*F_27(x))
Eq(F_45(x), F_26(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_15(x)*F_16(x)*F_20(x)*F_27(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_15(x)*F_16(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_57(x))
Eq(F_51(x), F_26(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_55(x))
Eq(F_53(x), F_16(x)*F_54(x))
Eq(F_54(x), F_26(x) + F_27(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_15(x)*F_16(x)*F_27(x))
Eq(F_57(x), F_26(x)*F_58(x))
Eq(F_58(x), F_37(x) + F_59(x))
Eq(F_59(x), F_39(x) + F_60(x) + F_61(x))
Eq(F_60(x), F_15(x)*F_58(x))
Eq(F_61(x), F_15(x)*F_62(x))
Eq(F_62(x), F_27(x) + F_59(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": [1, 0, 3, 2], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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