0123_0231_0312_2301
Counting sequence:
1, 1, 2, 6, 20, 65, 200, 583, 1621, 4329, 11170, 27984, 68350, 163325, 382960, 883417, 2009383, 4515269, 10040362, 22124650, 48371528, 105035337, 226720712, 486829835, 1040554073, 2215051537, 4698190786, 9932815428, 20938827730, 44024464677, 92343067792, 193275058205, 403728757675, 841815772429, 1752349244458, 3642135321374, 7559146030684, 15668044897489, 32435597918216, 67070214984783, 138538471686365, 285873030819833, 589338241224482, 1213860847079224, 2498090429752486, 5136918338013069, 10555311641474032, 21673573223524833, 44473046339284399, 91197892475684885, 186899384559994474, 382805968353573586, 783626335192808560, 1603281467377821593, 3278620528763575816, 6701356245569455891, 13690942867253170145, 27958346486768036769, 57069614478096520130, 116445071965355231276, 237501829949080784762, 484227031934953223989, 986900807943546295312, 2010695104034434843173, 4095177184363623294259, 8337928321316830013661, 16971004547812910792938, 34532304905984415377798, 70245201432686119626628, 142851586106806928033953, 290425538696483355190216, 590295810358705841527127, 1199481086648890090460005, 2436741105160737153975817, 4949040074047388426416034, 10049195875546605277255968, 20400623205996867607090766, 41405709321801049540462109, 84020344463216727973221104, 170458540565662713990673129, 345752784409783944350705719, 701176975376484921743720805, 1421696763866803909899852202, 2882079153961275952978106682, 5841529560377888172694058392, 11837801625666448879274062313, 23985088261154242826761330504, 48589146541951175790423383323, 98416233123187731855157549417, 199308346324946224259483160881, 403568452807033969617888334018, 817040425928350981434248311828, 1653887892485268047266111711010, 3347389866227668263328172140101, 6774007894969600864249009684112, 13706472114967730403684170370605, 27729856879992518157741518095547, 56093539060099151016230324462509, 113454728720426531433956220436138, 229444758641309521670904643599342, 463960119683531960947794820568236
Generating function in Maple syntax:
-(8*x^8-42*x^7+111*x^6-173*x^5+172*x^4-110*x^3+44*x^2-10*x+1)/(2*x-1)^2/(x-1)^7
Generating function in latex syntax:
-\frac{8 x^{8}-42 x^{7}+111 x^{6}-173 x^{5}+172 x^{4}-110 x^{3}+44 x^{2}-10 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{7}}
Generating function in sympy syntax:
(-8*x**8 + 42*x**7 - 111*x**6 + 173*x**5 - 172*x**4 + 110*x**3 - 44*x**2 + 10*x - 1)/((x - 1)**7*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^7*F(x)+8*x^8-42*x^7+111*x^6-173*x^5+172*x^4-110*x^3+44*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{7} F \! \left(x \right)+8 x^{8}-42 x^{7}+111 x^{6}-173 x^{5}+172 x^{4}-110 x^{3}+44 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
35+1151/60*n-34*2^n+23/144*n^4+13/16*n^3+1/240*n^5+1/720*n^6+4*2^n*n+1051/180*n^2
Explicit closed form in latex syntax:
35+\frac{1151 n}{60}-34 \,2^{n}+\frac{23 n^{4}}{144}+\frac{13 n^{3}}{16}+\frac{n^{5}}{240}+\frac{n^{6}}{720}+4 \,2^{n} n +\frac{1051 n^{2}}{180}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 200
a(7) = 583
a(8) = 1621
a(n+2) = 1/720*n^6-1/80*n^5+17/144*n^4-17/48*n^3+497/360*n^2-4*a(n)+4*a(n+1)-17/15*n+2, n >= 9
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) = 65
a \! \left(6\right) = 200
a \! \left(7\right) = 583
a \! \left(8\right) = 1621
a \! \left(n +2\right) = \frac{n^{6}}{720}-\frac{n^{5}}{80}+\frac{17 n^{4}}{144}-\frac{17 n^{3}}{48}+\frac{497 n^{2}}{360}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{17 n}{15}+2, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21151/
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[0,x]+F[5,x]
F[5,x] = F[31,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[15,x]*F[8,x]
F[8,x] = F[10,x]+F[9,x]
F[9,x] = F[1,x]+F[7,x]
F[10,x] = F[11,x]*F[15,x]
F[11,x] = F[10,x]+F[12,x]+F[16,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[15,x] = x
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[15,x]*F[18,x]
F[18,x] = F[19,x]+F[26,x]
F[19,x] = F[12,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[25,x]
F[22,x] = 0
F[23,x] = F[15,x]*F[24,x]
F[24,x] = F[13,x]+F[21,x]
F[25,x] = F[15,x]*F[20,x]
F[26,x] = F[13,x]+F[27,x]
F[27,x] = F[22,x]+F[28,x]+F[29,x]
F[28,x] = F[15,x]*F[26,x]
F[29,x] = F[15,x]*F[30,x]
F[30,x] = F[13,x]+F[27,x]
F[31,x] = F[32,x]
F[32,x] = F[15,x]*F[33,x]
F[33,x] = F[34,x]+F[5,x]
F[34,x] = F[35,x]*F[6,x]
F[35,x] = F[20,x]+F[36,x]
F[36,x] = F[12,x]+F[30,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_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{31}\! \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_{10}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{16}\! \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_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \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_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{15}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right) F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right)+F_{27}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{5}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{6}\! \left(x \right)
F_{35}\! \left(x \right) = F_{20}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_31(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_10(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_7(x))
Eq(F_10(x), F_11(x)*F_15(x))
Eq(F_11(x), F_10(x) + F_12(x) + F_16(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))
Eq(F_15(x), x)
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_15(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_26(x))
Eq(F_19(x), F_12(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_25(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_15(x)*F_24(x))
Eq(F_24(x), F_13(x) + F_21(x))
Eq(F_25(x), F_15(x)*F_20(x))
Eq(F_26(x), F_13(x) + F_27(x))
Eq(F_27(x), F_22(x) + F_28(x) + F_29(x))
Eq(F_28(x), F_15(x)*F_26(x))
Eq(F_29(x), F_15(x)*F_30(x))
Eq(F_30(x), F_13(x) + F_27(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_15(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_5(x))
Eq(F_34(x), F_35(x)*F_6(x))
Eq(F_35(x), F_20(x) + F_36(x))
Eq(F_36(x), F_12(x) + F_30(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, 2, 3], "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": [2, 3, 0, 1], "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, 2, 3], "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": [2, 3, 0, 1], "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, 2, 3], "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": [2, 3, 0, 1], "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, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 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": [[2, 1]]}, {"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, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [2, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": 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