0231_1203_2301
Counting sequence:
1, 1, 2, 6, 21, 75, 262, 894, 3011, 10120, 34213, 116864, 404013, 1413582, 5000943, 17866417, 64375380, 233666100, 853563171, 3135322766, 11572857733, 42900986647, 159646710614, 596141205139, 2233013001306, 8388100328056, 31590782697103, 119258534448656, 451201076279259, 1710531610501071, 6496944031161218, 24719956447461658, 94209816069698831, 359591013631701356, 1374498974880446873, 5260965198314033672, 20162189815793289133, 77362415561848561186, 297176045810492504987, 1142780339034784932222, 4398987259057253056071, 16949671798234789524162, 65368400693996381177479, 252320910054738650728488, 974764362383674445196849, 3768690714132539294356680, 14581774075289243340216245, 56460513887875730530296280, 218766287357910886933521805, 848212060282901117493554990, 3290827499792179523556148471, 12775305396954864635605161192, 49624082971971331126167700309, 192867419690423238041049525086, 749999759226169591406687830807, 2918034899675695005874501154416, 11358962148905938388359823567405, 44238235092686723711352254771250, 172369520560394758195972078406379, 671923229649159215654453409380832, 2620406438868302119182030339295861, 10223536120856781481402042300383306, 39903506296602086141196612830639787, 155809053412567111131354734448684261, 608612075190650770226350549394295032, 2378206052858151047846943797059796704, 9296392919029250722062058887823608895, 36352208502590672844544205539010979290, 142198149266264659567029643901944381297, 556415958263809600734248879842660708659, 2177928234263637494386302976218357295378, 8527504354990023951693345649689744371363, 33398806062549283782103000010265694602226, 130847946101821547129745636124811346463924, 512774443795677004145335326495398330104883, 2010045963542373699453686781913723631949876, 7881385175193925776132228182031180939905079, 30910995030584419868435702586689372730206583, 121264680647762468842913481692152123288327378, 475843053631008475507505442178453573356598371, 1867662797070294724979166769903505127771464942, 7332226895007186621009298812339935283629064166, 28792109207612567113524898809558629824423854457, 113085983737971757690455850574143925970790975526, 444262277944356620426777117238847128597709266477, 1745674173211168711819032427935461707209125790999, 6860848552155141566331125808437356144441237380270, 26970019681884040820917852633941111535425471102565, 106040419668385916084617289203582214106191888051232, 417009918676995336237239167178189244588238414288264, 1640227918774150618633257443771283177296311324157603, 6452723287921052611715037953254033717143482606486208, 25389908156754017175704795798567712098084243214811191, 99920993226623950827935309981510250091751933637740489, 393303863231779262500419881203371921762734605569938608, 1548367070060206133006492708699590666569933996990367961, 6096664861074148791093890410129894068382288302428121944, 24009432004599099955696336809569115895439266519024050338, 94567335185883987969154540483700566603643833317858298513, 372536385020076060518128477390757998217523952174400732726, 1467787353876267649128261326581606518013606363090473237897
Generating function in Maple syntax:
1/2*(-9*(x^2-x+1/3)^2*(1-4*x)^(1/2)-8*x^6+18*x^5-7*x^4-10*x^3+13*x^2-6*x+1)/x/(2*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{-9 \left(x^{2}-x +\frac{1}{3}\right)^{2} \sqrt{1-4 x}-8 x^{6}+18 x^{5}-7 x^{4}-10 x^{3}+13 x^{2}-6 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-4*x**6 + 9*x**5 - 7*x**4/2 - 5*x**3 + 13*x**2/2 - 3*x - 9*sqrt(1 - 4*x)*(x**2 - x + 1/3)**2/2 + 1/2)/(x*(x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^10*F(x)^2+(4*x^5-7*x^4+5*x^2-4*x+1)*(2*x-1)^2*(x-1)^5*F(x)+16*x^11-72*x^10+109*x^9+58*x^8-474*x^7+851*x^6-875*x^5+586*x^4-262*x^3+76*x^2-13*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{10} F \! \left(x \right)^{2}+\left(4 x^{5}-7 x^{4}+5 x^{2}-4 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+16 x^{11}-72 x^{10}+109 x^{9}+58 x^{8}-474 x^{7}+851 x^{6}-875 x^{5}+586 x^{4}-262 x^{3}+76 x^{2}-13 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(6) = 262
a(7) = 894
a(8) = 3011
a(n+5) = 12*(2*n+5)/(n+6)*a(n)-6*(11*n+25)/(n+6)*a(n+1)+(206+71*n)/(n+6)*a(n+2)-(38*n+147)/(n+6)*a(n+3)+(49+10*n)/(n+6)*a(n+4)+1/3*(2*n^3-24*n^2+19*n-57)/(n+6), 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) = 21
a \! \left(5\right) = 75
a \! \left(6\right) = 262
a \! \left(7\right) = 894
a \! \left(8\right) = 3011
a \! \left(n +5\right) = \frac{12 \left(2 n +5\right) a \! \left(n \right)}{n +6}-\frac{6 \left(11 n +25\right) a \! \left(n +1\right)}{n +6}+\frac{\left(206+71 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(38 n +147\right) a \! \left(n +3\right)}{n +6}+\frac{\left(49+10 n \right) a \! \left(n +4\right)}{n +6}+\frac{2 n^{3}-24 n^{2}+19 n -57}{3 n +18}, \quad n \geq 9
Specification 1
Strategy pack name: insertion_point_placements
Tree: http://permpal.com/tree/362/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[6,x]+F[9,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[6,x]^2*F[4,x]
F[9,x] = F[10,x]+F[2,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[4,x]
F[12,x] = F[13,x]+F[27,x]
F[13,x] = F[14,x]+F[23,x]
F[14,x] = F[15,x]+F[7,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]*F[6,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[6,x]*F[7,x]
F[19,x] = F[20,x]*F[22,x]*F[6,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[20,x]*F[6,x]
F[25,x] = F[26,x]
F[26,x] = F[22,x]*F[4,x]*F[6,x]*F[7,x]
F[27,x] = F[28,x]+F[56,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[20,x]*F[7,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[48,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[35,x]
F[35,x] = F[14,x]*F[20,x]*F[6,x]
F[36,x] = F[20,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[20,x]*F[6,x]
F[39,x] = F[40,x]+F[46,x]
F[40,x] = F[41,x]+F[42,x]+F[44,x]
F[41,x] = 0
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[20,x]+F[40,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[20,x]+F[40,x]
F[46,x] = F[47,x]
F[47,x] = F[20,x]^2*F[7,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[28,x]*F[6,x]
F[51,x] = F[20,x]^2*F[22,x]*F[6,x]
F[52,x] = F[53,x]+F[55,x]
F[53,x] = F[54,x]
F[54,x] = F[20,x]*F[28,x]*F[6,x]
F[55,x] = F[20,x]^2*F[37,x]
F[56,x] = F[10,x]+F[57,x]
F[57,x] = F[58,x]*F[7,x]
F[58,x] = F[41,x]+F[59,x]+F[61,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[20,x]+F[58,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[20,x]+F[58,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)+F_{9}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{6} \left(x \right)^{2} F_{4}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{2}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{27}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{6}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{22}\! \left(x \right) F_{6}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{20}\! \left(x \right) F_{6}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right) F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{56}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right) F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{48}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right) F_{20}\! \left(x \right) F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{20}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{20}\! \left(x \right) F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{46}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = 0
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{20}\! \left(x \right)+F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right)+F_{40}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{20} \left(x \right)^{2} F_{7}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{28}\! \left(x \right) F_{6}\! \left(x \right)
F_{51}\! \left(x \right) = F_{20} \left(x \right)^{2} F_{22}\! \left(x \right) F_{6}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{20}\! \left(x \right) F_{28}\! \left(x \right) F_{6}\! \left(x \right)
F_{55}\! \left(x \right) = F_{20} \left(x \right)^{2} F_{37}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right) F_{7}\! \left(x \right)
F_{58}\! \left(x \right) = F_{41}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{20}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{20}\! \left(x \right)+F_{58}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_6(x) + F_9(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_6(x)**2)
Eq(F_9(x), F_10(x) + F_2(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_4(x))
Eq(F_12(x), F_13(x) + F_27(x))
Eq(F_13(x), F_14(x) + F_23(x))
Eq(F_14(x), F_15(x) + F_7(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x)*F_6(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_6(x)*F_7(x))
Eq(F_19(x), F_20(x)*F_22(x)*F_6(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_20(x)*F_6(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_22(x)*F_4(x)*F_6(x)*F_7(x))
Eq(F_27(x), F_28(x) + F_56(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_20(x)*F_7(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_48(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_14(x)*F_20(x)*F_6(x))
Eq(F_36(x), F_20(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_20(x)*F_6(x))
Eq(F_39(x), F_40(x) + F_46(x))
Eq(F_40(x), F_41(x) + F_42(x) + F_44(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_20(x) + F_40(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_20(x) + F_40(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_20(x)**2*F_7(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_28(x)*F_6(x))
Eq(F_51(x), F_20(x)**2*F_22(x)*F_6(x))
Eq(F_52(x), F_53(x) + F_55(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_20(x)*F_28(x)*F_6(x))
Eq(F_55(x), F_20(x)**2*F_37(x))
Eq(F_56(x), F_10(x) + F_57(x))
Eq(F_57(x), F_58(x)*F_7(x))
Eq(F_58(x), F_41(x) + F_59(x) + F_61(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_20(x) + F_58(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_20(x) + F_58(x))
Pack JSON:
{"expansion_strats": [[{"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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_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:
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