0132_0231_0312_0321_1032_2301_3120
Counting sequence:
1, 1, 2, 6, 17, 38, 79, 160, 321, 642, 1283, 2564, 5125, 10246, 20487, 40968, 81929, 163850, 327691, 655372, 1310733, 2621454, 5242895, 10485776, 20971537, 41943058, 83886099, 167772180, 335544341, 671088662, 1342177303, 2684354584, 5368709145, 10737418266, 21474836507, 42949672988, 85899345949, 171798691870, 343597383711, 687194767392, 1374389534753, 2748779069474, 5497558138915, 10995116277796, 21990232555557, 43980465111078, 87960930222119, 175921860444200, 351843720888361, 703687441776682, 1407374883553323, 2814749767106604, 5629499534213165, 11258999068426286, 22517998136852527, 45035996273705008, 90071992547409969, 180143985094819890, 360287970189639731, 720575940379279412, 1441151880758558773, 2882303761517117494, 5764607523034234935, 11529215046068469816, 23058430092136939577, 46116860184273879098, 92233720368547758139, 184467440737095516220, 368934881474191032381, 737869762948382064702, 1475739525896764129343, 2951479051793528258624, 5902958103587056517185, 11805916207174113034306, 23611832414348226068547, 47223664828696452137028, 94447329657392904273989, 188894659314785808547910, 377789318629571617095751, 755578637259143234191432, 1511157274518286468382793, 3022314549036572936765514, 6044629098073145873530955, 12089258196146291747061836, 24178516392292583494123597, 48357032784585166988247118, 96714065569170333976494159, 193428131138340667952988240, 386856262276681335905976401, 773712524553362671811952722, 1547425049106725343623905363, 3094850098213450687247810644, 6189700196426901374495621205, 12379400392853802748991242326, 24758800785707605497982484567, 49517601571415210995964969048, 99035203142830421991929938009, 198070406285660843983859875930, 396140812571321687967719751771, 792281625142643375935439503452, 1584563250285286751870879006813
Generating function in Maple syntax:
(4*x^5-x^4-x^3-3*x^2+3*x-1)/(2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{4 x^{5}-x^{4}-x^{3}-3 x^{2}+3 x -1}{\left(2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(4*x**5 - x**4 - x**3 - 3*x**2 + 3*x - 1)/((x - 1)**2*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x-1)^2*F(x)+4*x^5-x^4-x^3-3*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+4 x^{5}-x^{4}-x^{3}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,-7+n+5/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ -7+n +\frac{5 \,2^{n}}{4} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 38
a(n+1) = 2*a(n)-n+8, 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) = 17
a \! \left(5\right) = 38
a \! \left(n +1\right) = 2 a \! \left(n \right)-n +8, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22932/
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[25,x]+F[8,x]
F[8,x] = F[22,x]+F[9,x]
F[9,x] = F[10,x]+F[15,x]
F[10,x] = F[11,x]*F[14,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[11,x]*F[13,x]
F[13,x] = x
F[14,x] = F[1,x]+F[13,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[17,x]*F[20,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[12,x]
F[19,x] = x^2
F[20,x] = 2*F[21,x]+F[1,x]
F[21,x] = F[13,x]*F[20,x]
F[22,x] = F[15,x]+F[23,x]
F[23,x] = F[19,x]+F[24,x]
F[24,x] = F[14,x]*F[18,x]
F[25,x] = F[26,x]+F[34,x]
F[26,x] = F[11,x]*F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[1,x]+F[30,x]+F[32,x]
F[30,x] = F[31,x]
F[31,x] = F[11,x]*F[13,x]*F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[11,x]*F[13,x]*F[14,x]
F[34,x] = F[13,x]*F[18,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_{25}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{22}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right) F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right)
F_{19}\! \left(x \right) = x^{2}
F_{20}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{1}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{34}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \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_25(x) + F_8(x))
Eq(F_8(x), F_22(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_15(x))
Eq(F_10(x), F_11(x)*F_14(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_11(x)*F_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_1(x) + F_13(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_17(x)*F_20(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_12(x))
Eq(F_19(x), x**2)
Eq(F_20(x), F_1(x) + 2*F_21(x))
Eq(F_21(x), F_13(x)*F_20(x))
Eq(F_22(x), F_15(x) + F_23(x))
Eq(F_23(x), F_19(x) + F_24(x))
Eq(F_24(x), F_14(x)*F_18(x))
Eq(F_25(x), F_26(x) + F_34(x))
Eq(F_26(x), F_11(x)*F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_1(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_11(x)*F_13(x)*F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_11(x)*F_13(x)*F_14(x))
Eq(F_34(x), F_13(x)*F_18(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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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, 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, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"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, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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": "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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], 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