0132_1302_3021_3120
Counting sequence:
1, 1, 2, 6, 20, 64, 191, 535, 1424, 3642, 9029, 21841, 51806, 120940, 278651, 635019, 1433756, 3211438, 7143617, 15794389, 34734314, 76022016, 165675287, 359661871, 778043720, 1677721954, 3607101821, 7734296985, 16542335414, 35299262932, 75161928179, 159719096851, 338765546036, 717259539030, 1516123456121, 3199750636189, 6743098655426, 14190571946728, 29824252904207, 62603443307319, 131254200566624, 274877906944906, 575044581327797, 1201766209160161, 2509085534585870, 5233675348214844, 10907155347539051, 22711512183342235, 47252611715302604, 98234766872020222, 204069358115226929, 423619839949538661, 878764877290669466, 1821706049271367120, 3774016487736477191, 7813745353487812159, 16167922662260082296, 33434723633598564018, 69103232682372892397, 142746093789135243049, 294715559615125260134, 608166093680111781796, 1254378597012249511907, 2586002934833157769251, 5328803194292846725220, 10975812723857183213734, 22597261490294200731881, 46504241809821779626285, 95664814766257734682994, 196716078802038658435512, 404352630095713371425279, 830841353079878204787271, 1706545187747018039102096, 3503995930289276748565210, 7192164153410469660461861, 14757395258967641292802929, 30270369155194425819794366, 62070785050838616688823308, 127239442514439720637828187, 260750187718130330119441579, 534194096542214266573294844, 1094077866751239403109092686, 2240139543745907860730547617, 4585455633798288459660529141, 9383682211848751654069341770, 19197742015480311294334078624, 39265910621083155594456600311, 80292017235525211267285389135, 164143112083995890824905756584, 335481750646338054497662668802, 685509296754279327225390043229, 1400419669441586435979634282681, 2860260460768871125154426523926, 5840601105348424136974068093300, 11923838458396782807828364530131, 24337901172350576204516782248499, 49666154376129456628952863871636, 101332819855744087782142712484598, 206706275999715656781556166431577, 421573052738400540335247359742909, 859625563279768062889951861150754
Generating function in Maple syntax:
(x^6-14*x^5+35*x^4-42*x^3+26*x^2-8*x+1)/(2*x-1)^3/(-1+x)^3
Generating function in latex syntax:
\frac{x^{6}-14 x^{5}+35 x^{4}-42 x^{3}+26 x^{2}-8 x +1}{\left(2 x -1\right)^{3} \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(x**6 - 14*x**5 + 35*x**4 - 42*x**3 + 26*x**2 - 8*x + 1)/((x - 1)**3*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^3*(-1+x)^3*F(x)-x^6+14*x^5-35*x^4+42*x^3-26*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{3} \left(-1+x \right)^{3} F \! \left(x \right)-x^{6}+14 x^{5}-35 x^{4}+42 x^{3}-26 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/16*(n^2+9*n-50)*2^n+1/2*n^2+3/2*n+4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(n^{2}+9 n -50\right) 2^{n}}{16}+\frac{n^{2}}{2}+\frac{3 n}{2}+4 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 191
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-1/2*(n-2)*(n-1), n >= 7
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) = 64
a \! \left(6\right) = 191
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-\frac{\left(n -2\right) \left(n -1\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21297/
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[8,x]
F[4,x] = F[40,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[26,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[8,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[8,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]+F[23,x]+F[25,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[8,x]
F[24,x] = F[15,x]+F[21,x]
F[25,x] = F[17,x]*F[8,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = F[22,x]+F[28,x]+F[30,x]
F[28,x] = F[29,x]*F[8,x]
F[29,x] = F[11,x]+F[27,x]
F[30,x] = F[31,x]*F[8,x]
F[31,x] = F[17,x]+F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[8,x]
F[35,x] = F[18,x]+F[33,x]
F[36,x] = 2*F[22,x]+F[37,x]+F[39,x]
F[37,x] = F[38,x]*F[8,x]
F[38,x] = F[21,x]+F[36,x]
F[39,x] = F[32,x]*F[8,x]
F[40,x] = F[41,x]
F[41,x] = F[42,x]*F[8,x]
F[42,x] = F[43,x]+F[51,x]
F[43,x] = F[4,x]+F[44,x]
F[44,x] = F[45,x]*F[6,x]
F[45,x] = F[14,x]+F[46,x]
F[46,x] = F[18,x]+F[47,x]
F[47,x] = F[22,x]+F[48,x]+F[50,x]
F[48,x] = F[49,x]*F[8,x]
F[49,x] = F[15,x]+F[47,x]
F[50,x] = F[46,x]*F[8,x]
F[51,x] = F[52,x]
F[52,x] = F[20,x]*F[53,x]
F[53,x] = F[54,x]+F[58,x]
F[54,x] = F[11,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[14,x]*F[20,x]
F[57,x] = F[50,x]
F[58,x] = F[14,x]*F[15,x]*F[59,x]
F[59,x] = F[28,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{40}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{26}\! \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_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{17}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{8}\! \left(x \right)
F_{38}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{32}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right) F_{8}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{51}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right) F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right) F_{8}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{46}\! \left(x \right) F_{8}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{11}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{14}\! \left(x \right) F_{20}\! \left(x \right)
F_{57}\! \left(x \right) = F_{50}\! \left(x \right)
F_{58}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{28}\! \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_8(x))
Eq(F_4(x), F_40(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_26(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_8(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_8(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_8(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_25(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_8(x))
Eq(F_24(x), F_15(x) + F_21(x))
Eq(F_25(x), F_17(x)*F_8(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), F_22(x) + F_28(x) + F_30(x))
Eq(F_28(x), F_29(x)*F_8(x))
Eq(F_29(x), F_11(x) + F_27(x))
Eq(F_30(x), F_31(x)*F_8(x))
Eq(F_31(x), F_17(x) + F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_8(x))
Eq(F_35(x), F_18(x) + F_33(x))
Eq(F_36(x), 2*F_22(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_38(x)*F_8(x))
Eq(F_38(x), F_21(x) + F_36(x))
Eq(F_39(x), F_32(x)*F_8(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_42(x)*F_8(x))
Eq(F_42(x), F_43(x) + F_51(x))
Eq(F_43(x), F_4(x) + F_44(x))
Eq(F_44(x), F_45(x)*F_6(x))
Eq(F_45(x), F_14(x) + F_46(x))
Eq(F_46(x), F_18(x) + F_47(x))
Eq(F_47(x), F_22(x) + F_48(x) + F_50(x))
Eq(F_48(x), F_49(x)*F_8(x))
Eq(F_49(x), F_15(x) + F_47(x))
Eq(F_50(x), F_46(x)*F_8(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_20(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_58(x))
Eq(F_54(x), F_11(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_14(x)*F_20(x))
Eq(F_57(x), F_50(x))
Eq(F_58(x), F_14(x)*F_15(x)*F_59(x))
Eq(F_59(x), F_28(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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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