0132_0231_0321_1320_3021_3120
Counting sequence:
1, 1, 2, 6, 18, 51, 137, 353, 881, 2145, 5121, 12033, 27905, 64001, 145409, 327681, 733185, 1630209, 3604481, 7929857, 17367041, 37879809, 82313217, 178257921, 384827393, 828375041, 1778384897, 3808428033, 8136949761, 17347641345, 36909875201, 78383153153, 166161547265, 351650447361, 743029342209, 1567663063041, 3302829850625, 6949257084929, 14602888806401, 30648886624257, 64252710748161, 134552735449089, 281474976710657, 588238720860161, 1228154488225793, 2561862092718081, 5339228464480257, 11118261580070913, 23133724648407041, 48097036645433345, 99923616732282881, 207447057835753473, 430375239390593025, 892275676172779521, 1848727647035588609, 3828059683264921601, 7921831744544702465, 16384095444373864449, 33867069197826129921, 69967923810828025857, 144475476046045511681, 298174324128945799169, 615083622707752861697, 1268213655067531673601, 2613673050943722094593, 5384143426513975377921, 11086493188299440521217, 22818622419178715348993, 46946963667590808862721, 96550258481795793158145, 198486966233114775388161, 407894404957865605332993, 837924902804182672605185, 1720712287195626974740481, 3532330129186494619844609, 7248832551204905403023361, 14870732054556512777928705, 30497042746372168790048769, 62524132233194102629335041, 128146136879150692518854657, 262563576447552273881497601, 537820874001058154097410049, 1101331421668927178157326337, 2254646653581283410827018241, 4614469853469039559853473793, 9441710651190253854455234561, 19313798894163315695105867777, 39498024378449164396000182273, 80756244750257228870372556801, 165071567113459926031080095745, 337338660705266124910011351041, 689223116872135468050087411713, 1407847309677298717629029023745, 2875115741240295688453216010241, 5870311666291273263571647070209, 11983259580282481061023522488321, 24456743416121972710907098169345, 49903838863672249641733495717889, 101808188830829673807703976181761, 207657013949886828832678693830657, 423474528638742884437492414545921
Generating function in Maple syntax:
(x^5-10*x^3+13*x^2-6*x+1)/(x-1)/(2*x-1)^3
Generating function in latex syntax:
\frac{x^{5}-10 x^{3}+13 x^{2}-6 x +1}{\left(x -1\right) \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(x**5 - 10*x**3 + 13*x**2 - 6*x + 1)/((x - 1)*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)^3*F(x)+x^5-10*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right)^{3} F \! \left(x \right)+x^{5}-10 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1+1/32*(n^2+7*n-10)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 1+\frac{\left(n^{2}+7 n -10\right) 2^{n}}{32} & \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+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-1, 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 +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22349/
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[10,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[10,x]*F[24,x]*F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[11,x]
F[10,x] = x
F[11,x] = F[12,x]+F[15,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[10,x]*F[12,x]
F[15,x] = F[16,x]+F[19,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[1,x]+F[16,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[10,x]*F[22,x]
F[22,x] = F[13,x]+F[19,x]
F[23,x] = F[10,x]*F[15,x]
F[24,x] = F[25,x]+F[47,x]
F[25,x] = F[26,x]+F[35,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[12,x]+F[30,x]
F[30,x] = F[16,x]+F[31,x]
F[31,x] = F[20,x]+F[32,x]+F[34,x]
F[32,x] = F[10,x]*F[33,x]
F[33,x] = F[13,x]+F[31,x]
F[34,x] = F[10,x]*F[30,x]
F[35,x] = F[16,x]+F[36,x]
F[36,x] = F[20,x]+F[32,x]+F[37,x]
F[37,x] = F[10,x]*F[38,x]
F[38,x] = F[30,x]+F[39,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[41,x]
F[41,x] = F[10,x]*F[42,x]
F[42,x] = F[16,x]+F[40,x]
F[43,x] = 2*F[20,x]+F[44,x]+F[46,x]
F[44,x] = F[10,x]*F[45,x]
F[45,x] = F[31,x]+F[43,x]
F[46,x] = F[10,x]*F[39,x]
F[47,x] = F[16,x]*F[26,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_{10}\! \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_{10}\! \left(x \right) F_{24}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \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_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \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_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{35}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{20}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{20}\! \left(x \right)+F_{32}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{10}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{16}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{31}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right) F_{26}\! \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_10(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_10(x)*F_24(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_15(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_10(x)*F_12(x))
Eq(F_15(x), F_16(x) + F_19(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_1(x) + F_16(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_13(x) + F_19(x))
Eq(F_23(x), F_10(x)*F_15(x))
Eq(F_24(x), F_25(x) + F_47(x))
Eq(F_25(x), F_26(x) + F_35(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_12(x) + F_30(x))
Eq(F_30(x), F_16(x) + F_31(x))
Eq(F_31(x), F_20(x) + F_32(x) + F_34(x))
Eq(F_32(x), F_10(x)*F_33(x))
Eq(F_33(x), F_13(x) + F_31(x))
Eq(F_34(x), F_10(x)*F_30(x))
Eq(F_35(x), F_16(x) + F_36(x))
Eq(F_36(x), F_20(x) + F_32(x) + F_37(x))
Eq(F_37(x), F_10(x)*F_38(x))
Eq(F_38(x), F_30(x) + F_39(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_10(x)*F_42(x))
Eq(F_42(x), F_16(x) + F_40(x))
Eq(F_43(x), 2*F_20(x) + F_44(x) + F_46(x))
Eq(F_44(x), F_10(x)*F_45(x))
Eq(F_45(x), F_31(x) + F_43(x))
Eq(F_46(x), F_10(x)*F_39(x))
Eq(F_47(x), F_16(x)*F_26(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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": "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, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], 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