021_1230_2310_3120
Counting sequence:
1, 1, 2, 5, 11, 22, 42, 79, 149, 284, 548, 1069, 2103, 4162, 8270, 16475, 32873, 65656, 131208, 262297, 524459, 1048766, 2097362, 4194535, 8388861, 16777492, 33554732, 67109189, 134218079, 268435834, 536871318, 1073742259, 2147484113, 4294967792, 8589935120, 17179869745, 34359738963, 68719477366, 137438954138, 274877907647, 549755814629, 1099511628556, 2199023256372, 4398046511965, 8796093023111, 17592186045362, 35184372089822, 70368744178699, 140737488356409, 281474976711784, 562949953422488, 1125899906843849, 2251799813686523, 4503599627371822, 9007199254742370, 18014398509483415, 36028797018965453, 72057594037929476, 144115188075857468, 288230376151713397, 576460752303425199, 1152921504606848746, 2305843009213695782, 4611686018427389795, 9223372036854777761, 18446744073709553632, 36893488147419105312, 73786976294838208609, 147573952589676415139, 295147905179352828134, 590295810358705654058, 1180591620717411305839, 2361183241434822609333, 4722366482869645216252, 9444732965739290430020, 18889465931478580857485, 37778931862957161712343, 75557863725914323421986, 151115727451828646841198, 302231454903657293679547, 604462909807314587356169, 1208925819614629174709336, 2417851639229258349415592, 4835703278458516698828025, 9671406556917033397652811, 19342813113834066795302302, 38685626227668133590601202, 77371252455336267181198919, 154742504910672534362394269, 309485009821345068724784884, 618970019642690137449566028, 1237940039285380274899128229, 2475880078570760549798252543, 4951760157141521099596501082, 9903520314283042199192998070, 19807040628566084398385991955, 39614081257132168796771979633, 79228162514264337593543954896, 158456325028528675187087905328, 316912650057057350374175806097, 633825300114114700748351607539
Generating function in Maple syntax:
-(x^4+3*x^3-6*x^2+4*x-1)/(2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{x^{4}+3 x^{3}-6 x^{2}+4 x -1}{\left(2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**4 - 3*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^3*F(x)+x^4+3*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{4}+3 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,2^(-1+n)-3/2*n+1/2*n^2+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 2^{-1+n}-\frac{3 n}{2}+\frac{n^{2}}{2}+1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 5
a(4) = 11
a(n+1) = 2*a(n)-1/2*(-1+n)*(n-4), n >= 5
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 5
a \! \left(4\right) = 11
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(-1+n \right) \left(n -4\right)}{2}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/5384/
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[5,x]+F[9,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]*F[8,x]
F[8,x] = x
F[9,x] = F[10,x]+F[21,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[8,x]
F[12,x] = F[13,x]+F[14,x]
F[13,x] = F[1,x]+F[8,x]
F[14,x] = F[15,x]+F[18,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[19,x]
F[19,x] = F[20,x]*F[8,x]
F[20,x] = F[18,x]+F[8,x]
F[21,x] = F[22,x]+F[23,x]+F[37,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[8,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]+F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[6,x]*F[8,x]
F[28,x] = F[29,x]+F[34,x]
F[29,x] = F[22,x]+F[30,x]+F[32,x]
F[30,x] = F[31,x]*F[8,x]
F[31,x] = F[29,x]+F[6,x]
F[32,x] = F[33,x]*F[8,x]
F[33,x] = F[15,x]+F[29,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[8,x]
F[36,x] = F[26,x]+F[34,x]
F[37,x] = F[8,x]*F[9,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_{5}\! \left(x \right)+F_{9}\! \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_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \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_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)+F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{37}\! \left(x \right)
F_{22}\! \left(x \right) = 0
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_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{6}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = F_{22}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right)+F_{6}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x \right) = F_{26}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{8}\! \left(x \right) F_{9}\! \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_5(x) + F_9(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_5(x)*F_8(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_21(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_8(x))
Eq(F_12(x), F_13(x) + F_14(x))
Eq(F_13(x), F_1(x) + F_8(x))
Eq(F_14(x), F_15(x) + F_18(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_19(x))
Eq(F_19(x), F_20(x)*F_8(x))
Eq(F_20(x), F_18(x) + F_8(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_37(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_8(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x) + F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_6(x)*F_8(x))
Eq(F_28(x), F_29(x) + F_34(x))
Eq(F_29(x), F_22(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_31(x)*F_8(x))
Eq(F_31(x), F_29(x) + F_6(x))
Eq(F_32(x), F_33(x)*F_8(x))
Eq(F_33(x), F_15(x) + F_29(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_8(x))
Eq(F_36(x), F_26(x) + F_34(x))
Eq(F_37(x), F_8(x)*F_9(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "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], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "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], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[2, 1], [2, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[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": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"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": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": 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