0132_0231_0321_1320_2031_3120
Counting sequence:
1, 1, 2, 6, 18, 50, 131, 329, 799, 1887, 4354, 9856, 21968, 48360, 105405, 227907, 489581, 1046061, 2225012, 4714546, 9956438, 20965326, 44032647, 92265581, 192927083, 402640235, 838845526, 1744812564, 3623857804, 7516168612, 15569228609, 32212222791, 66571956633, 137438912025, 283467794600, 584115499302, 1202590783346, 2473901095786, 5085241203947, 10445360380881, 21440476649463, 43980465008951, 90159953364842, 184717953342056, 378231999818248, 774056185804320, 1583296743833221, 3236962231993099, 6614661952504709, 13510798881898437, 27584547717412764, 56294995341880026, 114841790497675598, 234187180622971590, 477381560500954895, 972777519511684597, 1981583836042649411, 4035225266123567811, 8214565720323358782, 16717361816798824316, 34011184385901496388, 69175290276410294876, 140656423562034771337, 285924533142497452431, 581072438321850238513, 1180591620717410624305, 2398076729582240987216, 4869940435459320857934, 9887454823508318849514, 20070057552195991291362, 40730410914750689048819, 82641413450218790265561, 167644010141872404054863, 340010386766614454295055, 689465506498968200046290, 1397820478929414982035792, 2833419889721787126931776, 5742397643169488578498136, 11635911013790805805117517, 23574053482485268905264851, 47752569874777852399309213, 96714065569170333974827101, 195845982777569926300648132, 396527668833598369301784706, 802726744224113772002967878, 1624796301562061610803072190, 3288278229351791355198671511, 6653927711158918977580563069, 13462597927228510489525640219, 27234680864278366047778287387, 55088331748199422233008468774, 111414603535684224740918503396, 225305087149939210031637810428, 455561934457019941162874790420, 921027389228322924524945368785, 1861861819085211933448279644887, 3763337719427556035693334314441, 7605903601369376408980215762761, 15370263527767281493147522748152, 31057439705591620336669224762486, 62748704711297355374086804739938
Generating function in Maple syntax:
(2*x^7-6*x^6+24*x^5-47*x^4+48*x^3-27*x^2+8*x-1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
\frac{2 x^{7}-6 x^{6}+24 x^{5}-47 x^{4}+48 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(2*x**7 - 6*x**6 + 24*x**5 - 47*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)-2*x^7+6*x^6-24*x^5+47*x^4-48*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)-2 x^{7}+6 x^{6}-24 x^{5}+47 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1-1/24*n^4+1/4*n^3-23/24*n^2+3/4*n+2^(-1+n)*n-2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1-\frac{n^{4}}{24}+\frac{n^{3}}{4}-\frac{23 n^{2}}{24}+\frac{3 n}{4}+2^{-1+n} n -2^{-1+n} & \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) = 50
a(6) = 131
a(7) = 329
a(n+2) = -4*a(n)+4*a(n+1)-1/24*n*(n^3-14*n^2+59*n-94), 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) = 18
a \! \left(5\right) = 50
a \! \left(6\right) = 131
a \! \left(7\right) = 329
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{n \left(n^{3}-14 n^{2}+59 n -94\right)}{24}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22346/
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[12,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[27,x]+F[8,x]
F[8,x] = F[16,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]*F[9,x]
F[12,x] = x
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[16,x]
F[20,x] = F[21,x]+F[25,x]
F[21,x] = F[14,x]*F[22,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[22,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[13,x]*F[20,x]
F[27,x] = F[23,x]+F[28,x]
F[28,x] = F[29,x]+F[30,x]+F[33,x]
F[29,x] = 0
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[30,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[48,x]
F[35,x] = F[23,x]+F[36,x]
F[36,x] = F[29,x]+F[30,x]+F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[23,x]+F[40,x]
F[40,x] = F[29,x]+F[30,x]+F[41,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[43,x]+F[45,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[23,x]
F[45,x] = 2*F[29,x]+F[46,x]+F[47,x]
F[46,x] = F[12,x]*F[40,x]
F[47,x] = F[12,x]*F[42,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[23,x]+F[49,x]
F[52,x] = 2*F[29,x]+F[53,x]+F[55,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[36,x]+F[52,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[62,x]
F[57,x] = F[49,x]+F[58,x]
F[58,x] = 2*F[29,x]+F[59,x]+F[61,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[40,x]+F[58,x]
F[61,x] = F[12,x]*F[57,x]
F[62,x] = F[63,x]+F[65,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[49,x]
F[65,x] = 3*F[29,x]+F[66,x]+F[67,x]
F[66,x] = F[12,x]*F[58,x]
F[67,x] = F[12,x]*F[62,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_{12}\! \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_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{27}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \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_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{21}\! \left(x \right) = F_{14}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{48}\! \left(x \right)
F_{35}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{23}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \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_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{23}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{36}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{62}\! \left(x \right)
F_{57}\! \left(x \right) = F_{49}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{40}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{65}\! \left(x \right) = 3 F_{29}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \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_12(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_12(x)*F_7(x))
Eq(F_7(x), F_27(x) + F_8(x))
Eq(F_8(x), F_16(x)*F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_16(x))
Eq(F_20(x), F_21(x) + F_25(x))
Eq(F_21(x), F_14(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_22(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_13(x)*F_20(x))
Eq(F_27(x), F_23(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_33(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_30(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_48(x))
Eq(F_35(x), F_23(x) + F_36(x))
Eq(F_36(x), F_29(x) + F_30(x) + F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_23(x) + F_40(x))
Eq(F_40(x), F_29(x) + F_30(x) + F_41(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_43(x) + F_45(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_23(x))
Eq(F_45(x), 2*F_29(x) + F_46(x) + F_47(x))
Eq(F_46(x), F_12(x)*F_40(x))
Eq(F_47(x), F_12(x)*F_42(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_23(x) + F_49(x))
Eq(F_52(x), 2*F_29(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_36(x) + F_52(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_62(x))
Eq(F_57(x), F_49(x) + F_58(x))
Eq(F_58(x), 2*F_29(x) + F_59(x) + F_61(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_40(x) + F_58(x))
Eq(F_61(x), F_12(x)*F_57(x))
Eq(F_62(x), F_63(x) + F_65(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_49(x))
Eq(F_65(x), 3*F_29(x) + F_66(x) + F_67(x))
Eq(F_66(x), F_12(x)*F_58(x))
Eq(F_67(x), F_12(x)*F_62(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": [2, 0, 3, 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, 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