0132_0213_2301_3120
Counting sequence:
1, 1, 2, 6, 20, 58, 140, 296, 574, 1056, 1890, 3354, 5984, 10830, 19968, 37524, 71722, 138980, 272102, 536662, 1063772, 2115618, 4216532, 8415136, 16808630, 33591368, 67152010, 134267826, 268493304, 536937366, 1073817800, 2147570124, 4295065314, 8590045260, 17179993678, 34359877934, 68719632692, 137439127210, 274878099932, 549756027672, 1099511863982, 2199023515888, 4398046797362, 8796093336266, 17592186388240, 35184372464478, 70368744587280, 140737488801156, 281474977195034, 562949953946676, 1125899907411510, 2251799814300294, 4503599628034444, 9007199255456690, 18014398510252388, 36028797019792144, 72057594038817062, 144115188076809240, 288230376152732762, 576460752304515682, 1152921504608013992, 2305843009214939558, 4611686018428715992, 9223372036856190396, 18446744073711056850, 36893488147420703388, 73786976294839905950, 147573952589678216286, 295147905179354737764, 590295810358707676986, 1180591620717413447020, 2361183241434824873864, 4722366482869647609374, 9444732965739292957120, 18889465931478583524098, 37778931862957164524154, 75557863725914326384832, 151115727451828649961070, 302231454903657296962592, 604462909807314590808692, 1208925819614629178337802, 2417851639229258353226628, 4835703278458516702828422, 9671406556917033401849526, 19342813113834066799702460, 38685626227668133595212098, 77371252455336267186028020, 154742504910672534367449216, 309485009821345068730073494, 618970019642690137455096296, 1237940039285380274904908330, 2475880078570760549804290834, 4951760157141521099602806104, 9903520314283042199199578550, 19807040628566084398392856808, 39614081257132168796779137964, 79228162514264337593551416002, 158456325028528675187095678700, 316912650057057350374183901422, 633825300114114700748360034702, 1267650600228229401496711979412
Generating function in Maple syntax:
(6*x^7-5*x^6-8*x^5+13*x^4-18*x^3+15*x^2-6*x+1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{6 x^{7}-5 x^{6}-8 x^{5}+13 x^{4}-18 x^{3}+15 x^{2}-6 x +1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(6*x**7 - 5*x**6 - 8*x**5 + 13*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x-1)^5*F(x)+6*x^7-5*x^6-8*x^5+13*x^4-18*x^3+15*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+6 x^{7}-5 x^{6}-8 x^{5}+13 x^{4}-18 x^{3}+15 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,2^n+1/2*n^3-73/12*n^2+31/2*n-14+1/12*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2^{n}+\frac{n^{3}}{2}-\frac{73 n^{2}}{12}+\frac{31 n}{2}-14+\frac{n^{4}}{12} & \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) = 58
a(6) = 140
a(7) = 296
a(n+1) = -1/12*n^4-1/6*n^3+97/12*n^2+2*a(n)-155/6*n+24, 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) = 20
a \! \left(5\right) = 58
a \! \left(6\right) = 140
a \! \left(7\right) = 296
a \! \left(n +1\right) = -\frac{n^{4}}{12}-\frac{n^{3}}{6}+\frac{97 n^{2}}{12}+2 a \! \left(n \right)-\frac{155 n}{6}+24, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21201/
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[18,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[18,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[18,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[18,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[15,x]*F[18,x]
F[18,x] = x
F[19,x] = F[20,x]
F[20,x] = F[15,x]*F[18,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]
F[23,x] = F[18,x]*F[24,x]
F[24,x] = F[25,x]+F[39,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[15,x]^2
F[27,x] = F[28,x]
F[28,x] = F[15,x]*F[18,x]*F[29,x]*F[37,x]
F[29,x] = F[30,x]+F[34,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[15,x]^3
F[32,x] = F[16,x]*F[33,x]
F[33,x] = F[15,x]+F[16,x]
F[34,x] = F[35,x]
F[35,x] = F[15,x]*F[18,x]*F[36,x]
F[36,x] = F[13,x]
F[37,x] = F[1,x]+F[38,x]
F[38,x] = F[18,x]*F[37,x]
F[39,x] = F[40,x]
F[40,x] = F[15,x]*F[18,x]*F[37,x]*F[41,x]
F[41,x] = F[30,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[18,x]*F[44,x]
F[44,x] = F[19,x]+F[42,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_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \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_{18}\! \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_{18}\! \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_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = x
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \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_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15} \left(x \right)^{2}
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{29}\! \left(x \right) F_{37}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15} \left(x \right)^{3}
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right)
F_{37}\! \left(x \right) = F_{1}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right) F_{37}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{37}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{30}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{19}\! \left(x \right)+F_{42}\! \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_18(x)*F_4(x))
Eq(F_4(x), F_22(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_18(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_18(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_18(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_15(x)*F_18(x))
Eq(F_18(x), x)
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_15(x)*F_18(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_18(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_39(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_15(x)**2)
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_15(x)*F_18(x)*F_29(x)*F_37(x))
Eq(F_29(x), F_30(x) + F_34(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_15(x)**3)
Eq(F_32(x), F_16(x)*F_33(x))
Eq(F_33(x), F_15(x) + F_16(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_15(x)*F_18(x)*F_36(x))
Eq(F_36(x), F_13(x))
Eq(F_37(x), F_1(x) + F_38(x))
Eq(F_38(x), F_18(x)*F_37(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_15(x)*F_18(x)*F_37(x)*F_41(x))
Eq(F_41(x), F_30(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_18(x)*F_44(x))
Eq(F_44(x), F_19(x) + F_42(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": [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": [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": "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": [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": []}, "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, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 1], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": 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