0123_0132_0231_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 55, 149, 385, 961, 2340, 5596, 13203, 30828, 71391, 164230, 375736, 855707, 1941269, 4389449, 9896867, 22259425, 49957026, 111908240, 250270397, 558886636, 1246458957, 2776742994, 6179481762, 13739640507, 30524459787, 67765272597, 150343927901, 333359799641, 738779839608, 1636493542172, 3623526753663, 8020193085828, 17745599191123, 39252164089974, 86799399173348, 191894153352371, 424139982423193, 937278387276025, 2070848974416303, 4574634024278825, 10104138621878950, 22314310590263936, 49273623948985305, 108792124008205876, 240180033583387377, 530196641069182450, 1170311306053414526, 2583054445503902851, 5700809131704359871, 12580936768716876533, 27762942381447139225, 61262722691617603665, 135178439745990013228, 298263937061503023036, 658078827190775363019, 1451912554879844164348, 3203241968325798200359, 7066868606851585459398, 15590261454601442472816, 34392988249565537923627, 75871291850056370860157, 167369738642861603298313, 369206252511583582728731, 814431370825813212732625, 1796527628199667381591562, 3962851804721277051565776, 8741315571889084727169877, 19281519955219271658540508, 42530614081642690013862901, 93811988468140204045325554, 206924386357431158330048954, 456417165728367963835673003, 1006721877788602046040093395, 2220519257662087079057076789, 4897757912507445779243505973, 10802842165713300919114461353, 23827412514908303546460708672, 52555000793963282130514338748, 115917679456918323696841966711, 255672442835301867973542294740, 563919229277680852144394230363, 1243794823638507696119221028470, 2743339461364772596479165550428, 6050752894512136717637087725315, 13345610097672602476462121263809, 29435178626729620239540857643897, 64922348088010662576993702141159, 143192782153772498390999323798489, 315825694694431758542639101741774, 696583640997188462704471098622592, 1536379871188777989884339907035441, 3388625051153244870480115687792100, 7473912971466192468002296018161497, 16484364270446191454564119031263570, 36357670504695684836958727926125142, 80189887806157676256620500222019123
Generating function in Maple syntax:
(2*x^7-5*x^6+8*x^5-10*x^4+15*x^3-14*x^2+6*x-1)/(2*x-1)/(x^3+2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{2 x^{7}-5 x^{6}+8 x^{5}-10 x^{4}+15 x^{3}-14 x^{2}+6 x -1}{\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(2*x**7 - 5*x**6 + 8*x**5 - 10*x**4 + 15*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^3+2*x-1)*(x-1)^3*F(x)-2*x^7+5*x^6-8*x^5+10*x^4-15*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-2 x^{7}+5 x^{6}-8 x^{5}+10 x^{4}-15 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
5/8-1/4*n^2+1/2*n-6909/236*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+351/472*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-27535/472*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+14463/472*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{5}{8}-\frac{n^{2}}{4}+\frac{n}{2}-\frac{6909 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{236}+\frac{351 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{472}-\frac{27535 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{472}+\frac{14463 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{472}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 149
a(7) = 385
a(n+4) = -2*a(n)+a(n+1)-4*a(n+2)+4*a(n+3)-1/2*(n+1)*(n-4), n >= 8
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) = 55
a \! \left(6\right) = 149
a \! \left(7\right) = 385
a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)-\frac{\left(n +1\right) \left(n -4\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16158/
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[17,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]
F[15,x] = F[4,x]
F[16,x] = F[12,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[45,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[24,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[13,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[20,x]
F[26,x] = 2*F[19,x]+F[27,x]+F[31,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[23,x]
F[30,x] = F[27,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[44,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[12,x]+F[4,x]
F[38,x] = F[34,x]+F[39,x]
F[39,x] = 2*F[19,x]+F[31,x]+F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[12,x]
F[43,x] = F[40,x]
F[44,x] = F[39,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[2,x]+F[48,x]
F[48,x] = F[19,x]+F[35,x]+F[49,x]
F[49,x] = F[2,x]*F[4,x]
F[50,x] = F[18,x]+F[51,x]
F[51,x] = 2*F[19,x]+F[31,x]+F[40,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_{17}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{45}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \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_{44}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{31}\! \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_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{2}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{31}\! \left(x \right)+F_{40}\! \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_17(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_12(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_45(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_24(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_13(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_20(x))
Eq(F_26(x), 2*F_19(x) + F_27(x) + F_31(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_23(x))
Eq(F_30(x), F_27(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_44(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_12(x) + F_4(x))
Eq(F_38(x), F_34(x) + F_39(x))
Eq(F_39(x), 2*F_19(x) + F_31(x) + F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_12(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_39(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_2(x) + F_48(x))
Eq(F_48(x), F_19(x) + F_35(x) + F_49(x))
Eq(F_49(x), F_2(x)*F_4(x))
Eq(F_50(x), F_18(x) + F_51(x))
Eq(F_51(x), 2*F_19(x) + F_31(x) + F_40(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, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 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, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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