0213_0231_1032_2301
Counting sequence:
1, 1, 2, 6, 20, 62, 176, 469, 1202, 3014, 7475, 18457, 45550, 112619, 279355, 695835, 1741323, 4378945, 11065553, 28093874, 71639461, 183411941, 471250512, 1214603826, 3138997885, 8131076262, 21103223556, 54859812624, 142805108422, 372146206344, 970680697190, 2533721842707, 6617599897980, 17292234270104, 45203268297025, 118201606458095, 309161032856848, 808783275847765, 2116158002469953, 5537560427710089, 14492125234067621, 37929744303472931, 99278415978578447, 259867020725179096, 680243481359636375, 1780700695632684287, 4661524354003424486, 12203186271121694704, 31946627084477928967, 83633809863800614424, 218948891532412779730, 573200761309438937382, 1500628622597953245960, 3928634440988612612914, 10285171117576454779820, 26926667242558264998305, 70494398264534112307798, 184555644845517106942010, 483170734832166259926303, 1264952884713685738089429, 3311680425319111009383106, 8670073115033711248851471, 22698507790901398351782807, 59425386846987730429357191, 155577523622853276968833615, 407306921155469050115399637, 1066342704887975735799693341, 2791720105150557808419416334, 7308815396954408844311597337, 19134721584707114739383969129, 50095340207581874813901846220, 131151280443720483403092625982, 343358463344647712438213352865, 898924032851767307279811683674, 2353413479372560274702928560240, 6161316088867359164562743531340, 16130534144987675548713551758298, 42230285042722518209555831483180, 110560318338654648672952621670978, 289450664608633103269385069347519, 757791664606912284603540012516416, 1983924307149207542574252528235436, 5193981212110455017377938106400717, 13598019238512721038462373686257611, 35600076319670983516585548395198980, 93202209348151077069988485687910937, 244006550970412536253851303649858621, 638817442035004295698674148430938389, 1672445772039750252628720454392923657, 4378519867817175013305249573068687959, 11463113818722889384611880447094679035, 30010821562664237325358751064055970132, 78569350817276340941478401199307345195, 205697230783939882159819129167437716259, 538522341321617618780893579020353521410, 1409869792750109840511549272228724572700, 3691087036057202115096846524136833563723, 9663391313658669888836608788725419905900, 25299086901353540238271084650329944928478, 66233869383192188037178590441251911791234, 173402521233645041970640048556213699823676
Generating function in Maple syntax:
-(11*x^8-62*x^7+150*x^6-225*x^5+220*x^4-137*x^3+52*x^2-11*x+1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
-\frac{11 x^{8}-62 x^{7}+150 x^{6}-225 x^{5}+220 x^{4}-137 x^{3}+52 x^{2}-11 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-11*x**8 + 62*x**7 - 150*x**6 + 225*x**5 - 220*x**4 + 137*x**3 - 52*x**2 + 11*x - 1)/((x - 1)**5*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*(x-1)^5*F(x)+11*x^8-62*x^7+150*x^6-225*x^5+220*x^4-137*x^3+52*x^2-11*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+11 x^{8}-62 x^{7}+150 x^{6}-225 x^{5}+220 x^{4}-137 x^{3}+52 x^{2}-11 x +1 = 0
Explicit closed form in Maple syntax:
1/120*(-12*5^(1/2)+60)*(3/2-1/2*5^(1/2))^(-n)+1/120*(12*5^(1/2)+60)*(3/2+1/2*5^(1/2))^(-n)+1/24*n^4-1/4*n^3+11/24*n^2+2^(-1+n)*n+3/4*n-2^(n+1)+2
Explicit closed form in latex syntax:
\frac{\left(-12 \sqrt{5}+60\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(12 \sqrt{5}+60\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{4}+\frac{11 n^{2}}{24}+2^{-1+n} n +\frac{3 n}{4}-2^{n +1}+2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 176
a(7) = 469
a(8) = 1202
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-1/24*n*(n^3-10*n^2-n+106), 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) = 62
a \! \left(6\right) = 176
a \! \left(7\right) = 469
a \! \left(8\right) = 1202
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{n \left(n^{3}-10 n^{2}-n +106\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21322/
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[41,x]+F[8,x]
F[8,x] = F[20,x]+F[9,x]
F[9,x] = F[10,x]*F[17,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[14,x]
F[13,x] = x
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[14,x]*F[17,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[13,x]*F[17,x]
F[20,x] = F[21,x]*F[24,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[13,x]*F[21,x]
F[24,x] = F[25,x]+F[30,x]
F[25,x] = F[26,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[25,x]
F[29,x] = F[25,x]
F[30,x] = F[31,x]+F[32,x]+F[37,x]
F[31,x] = 0
F[32,x] = F[13,x]*F[33,x]
F[33,x] = F[22,x]+F[34,x]
F[34,x] = F[31,x]+F[32,x]+F[35,x]
F[35,x] = F[13,x]*F[36,x]
F[36,x] = F[18,x]+F[34,x]
F[37,x] = F[13,x]*F[38,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[18,x]+F[30,x]
F[40,x] = F[30,x]
F[41,x] = F[42,x]
F[42,x] = F[17,x]*F[43,x]
F[43,x] = F[44,x]+F[46,x]
F[44,x] = F[11,x]+F[45,x]
F[45,x] = F[11,x]*F[18,x]
F[46,x] = F[18,x]^2*F[21,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_{41}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{20}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \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_{14}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{37}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{22}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{40}\! \left(x \right) = F_{30}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right) F_{18}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18} \left(x \right)^{2} F_{21}\! \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_41(x) + F_8(x))
Eq(F_8(x), F_20(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_17(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_14(x))
Eq(F_13(x), x)
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_14(x)*F_17(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_13(x)*F_17(x))
Eq(F_20(x), F_21(x)*F_24(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_13(x)*F_21(x))
Eq(F_24(x), F_25(x) + F_30(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_25(x))
Eq(F_29(x), F_25(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_37(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_13(x)*F_33(x))
Eq(F_33(x), F_22(x) + F_34(x))
Eq(F_34(x), F_31(x) + F_32(x) + F_35(x))
Eq(F_35(x), F_13(x)*F_36(x))
Eq(F_36(x), F_18(x) + F_34(x))
Eq(F_37(x), F_13(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_18(x) + F_30(x))
Eq(F_40(x), F_30(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_17(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_46(x))
Eq(F_44(x), F_11(x) + F_45(x))
Eq(F_45(x), F_11(x)*F_18(x))
Eq(F_46(x), F_18(x)**2*F_21(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, 2, 1, 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": [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]]}], "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, 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": [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]]}], "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, 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": [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]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 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