0132_0231_1032_2031_3021_3120
Counting sequence:
1, 1, 2, 6, 18, 51, 136, 345, 842, 1995, 4620, 10509, 23566, 52239, 114704, 249873, 540690, 1163283, 2490388, 5308437, 11272214, 23855127, 50331672, 105906201, 222298138, 465567771, 973078556, 2030043165, 4227858462, 8791261215, 18253611040, 37849399329, 78383153186, 162135015459, 335007449124, 691489734693, 1425929142310, 2937757630503, 6047313952808, 12438225289257, 25563645345834, 52501680226347, 107752139522092, 221001837183021, 452998790643758, 927987813842991, 1899956092796976, 3887873115815985, 7951668092076082, 16255179905040435, 33214047251857460, 67835469387268149, 138485688541642806, 282600876617498679, 576460752303423544, 1175439502743699513, 2395915001761103930, 4881901996069617723, 9943947977234055228, 20248183924657750077, 41216943789694779454, 83875039460148117567, 170632382681813352512, 347029372886660939841, 705587960819390349378, 1434234351730917638211, 2914585563646109155396, 5921404847660766068805, 12027277136058627653702, 24423489153591446339655, 49584848070131274743880, 100645435666159313616969, 204242350384112155492426, 414387658871811367501899, 840581233950796848037964, 1704774300315941922144333, 3456772265460580296425550, 7007991860578553497124943, 14204878380471892802797648, 28787546079573357222690897, 58330670796405857679573074, 118172498867330001827528787, 239367312283696576591822932, 484779253665466299057176661, 981647765527078889861414998, 1987474047446450363216953431, 4023305127677485893422153816, 8143324320924142120820801625, 16480076772986624909594591322, 33347009808249931155095158875, 67467732141053224982002270300, 136482889331213175307628445789, 276060628760639801302504702046, 558310957717706503979505025119, 1129001315828266810708001292384, 2282761432442241226913985069153, 4615040466455897664823935107170, 9329116136054625751639800152163, 18856302678394912347263460180068, 38108746169361146382494640111717, 77009773963864936140924719726694
Generating function in Maple syntax:
(x^5-5*x^3+9*x^2-5*x+1)/(2*x-1)^2/(x-1)^2
Generating function in latex syntax:
\frac{x^{5}-5 x^{3}+9 x^{2}-5 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**5 - 5*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^2*F(x)+x^5-5*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+x^{5}-5 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/8*(5*n-14)*2^n+n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(5 n -14\right) 2^{n}}{8}+n +2 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 51
a(n+2) = -4*a(n)+4*a(n+1)+n, n >= 6
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) = 18
a \! \left(5\right) = 51
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+n, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22377/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[31,x]+F[8,x]
F[8,x] = F[19,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[9,x]
F[13,x] = x
F[14,x] = F[11,x]+F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[13,x]*F[18,x]
F[18,x] = F[1,x]+F[16,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[13,x]*F[22,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[30,x]
F[27,x] = 0
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[23,x]+F[26,x]
F[30,x] = F[13,x]*F[25,x]
F[31,x] = F[18,x]*F[32,x]
F[32,x] = F[13,x]+F[33,x]
F[33,x] = F[27,x]+F[34,x]+F[36,x]
F[34,x] = F[13,x]*F[35,x]
F[35,x] = F[20,x]
F[36,x] = F[13,x]*F[37,x]
F[37,x] = F[38,x]+F[42,x]
F[38,x] = F[13,x]+F[39,x]
F[39,x] = F[27,x]+F[40,x]+F[41,x]
F[40,x] = F[13,x]*F[23,x]
F[41,x] = F[13,x]*F[38,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[15,x]+F[27,x]+F[44,x]
F[44,x] = F[13,x]*F[45,x]
F[45,x] = F[13,x]+F[43,x]
F[46,x] = F[27,x]+F[47,x]+F[48,x]+F[50,x]
F[47,x] = F[13,x]*F[26,x]
F[48,x] = F[13,x]*F[49,x]
F[49,x] = F[39,x]+F[46,x]
F[50,x] = F[13,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{31}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{19}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{11}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \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_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{20}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{27}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{27}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{13}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{27}\! \left(x \right)+F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{26}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_31(x) + F_8(x))
Eq(F_8(x), F_19(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_9(x))
Eq(F_13(x), x)
Eq(F_14(x), F_11(x) + F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_13(x)*F_18(x))
Eq(F_18(x), F_1(x) + F_16(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_13(x)*F_22(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_30(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_23(x) + F_26(x))
Eq(F_30(x), F_13(x)*F_25(x))
Eq(F_31(x), F_18(x)*F_32(x))
Eq(F_32(x), F_13(x) + F_33(x))
Eq(F_33(x), F_27(x) + F_34(x) + F_36(x))
Eq(F_34(x), F_13(x)*F_35(x))
Eq(F_35(x), F_20(x))
Eq(F_36(x), F_13(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_42(x))
Eq(F_38(x), F_13(x) + F_39(x))
Eq(F_39(x), F_27(x) + F_40(x) + F_41(x))
Eq(F_40(x), F_13(x)*F_23(x))
Eq(F_41(x), F_13(x)*F_38(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_15(x) + F_27(x) + F_44(x))
Eq(F_44(x), F_13(x)*F_45(x))
Eq(F_45(x), F_13(x) + F_43(x))
Eq(F_46(x), F_27(x) + F_47(x) + F_48(x) + F_50(x))
Eq(F_47(x), F_13(x)*F_26(x))
Eq(F_48(x), F_13(x)*F_49(x))
Eq(F_49(x), F_39(x) + F_46(x))
Eq(F_50(x), F_13(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, 3, 1], "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, 0, 3, 1], "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, 3, 1], "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, 0, 3, 1], "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, 3, 1], "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, 0, 3, 1], "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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": 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