0123_0213_0231_1032_1230
Counting sequence:
1, 1, 2, 6, 19, 56, 158, 435, 1180, 3170, 8459, 22464, 59446, 156899, 413300, 1087098, 2856187, 7497848, 19670126, 51578067, 135195148, 354269522, 928137707, 2431192176, 6367535974, 16675610051, 43667682788, 114344215530, 299398518235, 783918448040, 2052491043614, 5373823118259, 14069515182076, 36835796169794, 96440020810955, 252488561230368, 661034252814742, 1730631377083043, 4530894238172756, 11862120056911962, 31055603371516603, 81304964935544792, 212859841190931662, 557275658148877971, 1458969332278957804, 3819636736734506546, 9999949674017584043, 26180229877504290000, 68540775142867374790, 179442165919842012035, 469785863354147016644, 1229915705617575748554, 3219961816448533650331, 8429970869627932045064, 22069953044235076170110, 57779892766676923835763, 151269734262994950078172, 396029328036706435880738, 1036818285875921376528011, 2714425601648651731631232, 7106458663185221894221558, 18604950676137390102745187, 48708393941687700717437492, 127520232301847216656414266, 333852305269696958465499259, 874036688118929677167471416, 2288257768310464109891690798, 5990736635259206726217152595, 15683952174360644216178870220, 41061119961609702217157664530, 107499407858042415024970536299, 281437103907665448037106770224, 736811904455249739445055426086, 1928998610638675391015470811459, 5050183929821959675036179615140, 13221553183549570116962713247658, 34614475630271483641591250555227, 90621873726154346739289619272808, 237251145585970488439234768972766, 621131563107314982304329011064627, 1626143543887090185925580911059388, 4257299068856187030376071015790082, 11145753663285933815009946723663947, 29179961922210540234268398329907936, 76394132105763538527024506615472214, 200002434399915778625263638215333411, 523613171103655203905683441428177428, 1370837078930392646205620752864497690, 3588898065726208360938846950755913275, 9395857118325603689066256366584437400, 24598673289405345211170594683359789454, 64400162750199916954266872752219712019, 168601814961813375671272713710748908716, 441405282136478150098836648654926138354, 1155614031450096954703807992803827754795, 3025436812218764474169728850856153622928, 7920696405216099988119661601963826107782, 20736652403449342530817822039433710688003, 54289260805171541685590936685134077931396, 142131130012144510688469252353562067056522, 372104129231420446704845349050739211138843
Generating function in Maple syntax:
-(2*x^5+3*x^3-8*x^2+5*x-1)/(x-1)/(2*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
-\frac{2 x^{5}+3 x^{3}-8 x^{2}+5 x -1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(-2*x**5 - 3*x**3 + 8*x**2 - 5*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)*(x^2-3*x+1)*F(x)+2*x^5+3*x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+2 x^{5}+3 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,-2/5*(3/2-1/2*5^(1/2))^(-n)-2/5*(3/2+1/2*5^(1/2))^(-n)+13/5*(3/2-1/2*5^(1/2))^(1-n)-2^(-1+n)+13/5*(3/2+1/2*5^(1/2))^(1-n)-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ -\frac{2 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{5}-\frac{2 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5}+\frac{13 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}}{5}-2^{-1+n}+\\\frac{13 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}}{5}-1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(n+3) = 2*a(n)-7*a(n+1)+5*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) = 19
a \! \left(5\right) = 56
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16704/
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[14,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[65,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[31,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[1,x]+F[4,x]
F[23,x] = F[16,x]+F[24,x]+F[29,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]+F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[11,x]*F[4,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[7,x]
F[31,x] = F[32,x]+F[55,x]
F[32,x] = F[16,x]+F[17,x]+F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[42,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[4,x]
F[42,x] = F[35,x]+F[43,x]
F[43,x] = 2*F[16,x]+F[44,x]+F[53,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[4,x]
F[49,x] = F[50,x]
F[50,x] = 2*F[16,x]+F[44,x]+F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[35,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[35,x]
F[55,x] = F[16,x]+F[56,x]+F[61,x]+F[63,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[47,x]
F[59,x] = F[60,x]
F[60,x] = 2*F[16,x]+F[51,x]+F[56,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[35,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[15,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[69,x]
F[67,x] = F[2,x]+F[68,x]
F[68,x] = F[36,x]
F[69,x] = F[15,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_{14}\! \left(x \right)+F_{6}\! \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_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{65}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{31}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \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_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{7}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{55}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{35}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{35}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right)+F_{56}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{47}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{35}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{69}\! \left(x \right)
F_{67}\! \left(x \right) = F_{2}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{36}\! \left(x \right)
F_{69}\! \left(x \right) = F_{15}\! \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_14(x) + F_6(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_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_65(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_31(x))
Eq(F_19(x), F_20(x) + F_23(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_4(x))
Eq(F_23(x), F_16(x) + F_24(x) + F_29(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x) + F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_11(x)*F_4(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_7(x))
Eq(F_31(x), F_32(x) + F_55(x))
Eq(F_32(x), F_16(x) + F_17(x) + F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_42(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_4(x))
Eq(F_42(x), F_35(x) + F_43(x))
Eq(F_43(x), 2*F_16(x) + F_44(x) + F_53(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_4(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), 2*F_16(x) + F_44(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_35(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_35(x))
Eq(F_55(x), F_16(x) + F_56(x) + F_61(x) + F_63(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_47(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), 2*F_16(x) + F_51(x) + F_56(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_35(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_15(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_69(x))
Eq(F_67(x), F_2(x) + F_68(x))
Eq(F_68(x), F_36(x))
Eq(F_69(x), F_15(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "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, 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": [1, 2, 3, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [1, 2, 3, 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