0132_0231_1032_1302_2301_3120
Counting sequence:
1, 1, 2, 6, 18, 46, 107, 237, 512, 1092, 2313, 4879, 10262, 21534, 45095, 94257, 196668, 409672, 852053, 1769571, 3670130, 7602306, 15728787, 32506021, 67109048, 138412236, 285212897, 587202807, 1207959822, 2483028262, 5100273983, 10468983129, 21474836852, 44023415184, 90194313645, 184683594187, 377957122538, 773094113802, 1580547965483, 3229815407181, 6597069767280, 13469017440916, 27487790695097, 56075093017311, 114349209289478, 233096465089326, 474989023200087, 967570232443777, 1970324836975532, 4011018418127832, 8162774324610053, 16607023625929779, 33776997205279842, 68679894317401234, 139611588448486595, 283726776524342517, 576460752303424808, 1170935903116330332, 2377900603251623313, 4827858800541173191, 9799832789158200830, 19887895954468111926, 40352252661239645807, 81857426827086137001, 166020696663385966308, 336653079345199318816, 682529530727253411677, 1383505805528216373147, 2803905099203851847642, 5681597174702541899802, 11510768301994760210523, 23316684509168873244829, 47223664828696452139232, 95627921278110315579684, 193617025797655453763945, 391956418078180552739247, 793357569122100395903478, 1605604604175679372659262, 3248988140214315907025543, 6573534144154546137467601, 13298184015760920921770780, 26898599486425499137215336, 54401661882658312861780917, 110012249584931254898265091, 222442350809091768145939538, 449720404896642052990700706, 909112216350201139379047667, 1837567245814236345553390917, 3713820117856140824697376152, 7505011488167617916575944172, 15164765481245908367514275393, 30639015972313161803753328279, 61897001964269013744956215022, 125031943967823407764811550534, 252539768014217576079421345695, 510031296185576673258439184377, 1029966112685436388716071358548, 2079739265999438861830528700592, 4199092613256009892457829372173, 8477413389026284122509202690411, 17113283103081096920205493277130
Generating function in Maple syntax:
-(5*x^6-4*x^5+5*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{5 x^{6}-4 x^{5}+5 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-5*x**6 + 4*x**5 - 5*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^3*F(x)+5*x^6-4*x^5+5*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+5 x^{6}-4 x^{5}+5 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,4-9/2*n+1/2*n^2+1/8*2^n*n+2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 4-\frac{9 n}{2}+\frac{n^{2}}{2}+\frac{2^{n} n}{8}+2^{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) = 46
a(6) = 107
a(n+2) = 1/2*n^2-4*a(n)+4*a(n+1)-13/2*n+13, 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) = 18
a \! \left(5\right) = 46
a \! \left(6\right) = 107
a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{13 n}{2}+13, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22368/
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[41,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[24,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[22,x] = F[23,x]*F[8,x]
F[23,x] = F[15,x]+F[21,x]
F[24,x] = F[18,x]+F[25,x]
F[25,x] = F[26,x]+F[27,x]+F[32,x]
F[26,x] = 0
F[27,x] = F[28,x]*F[8,x]
F[28,x] = F[15,x]+F[29,x]
F[29,x] = F[26,x]+F[27,x]+F[30,x]
F[30,x] = F[31,x]*F[8,x]
F[31,x] = F[18,x]+F[29,x]
F[32,x] = F[33,x]*F[8,x]
F[33,x] = F[31,x]+F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[8,x]
F[37,x] = F[18,x]+F[35,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]*F[8,x]
F[40,x] = F[29,x]+F[38,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]*F[8,x]
F[43,x] = F[44,x]+F[56,x]
F[44,x] = F[45,x]+F[52,x]
F[45,x] = F[20,x]*F[46,x]
F[46,x] = F[14,x]+F[47,x]
F[47,x] = F[18,x]+F[48,x]
F[48,x] = F[26,x]+F[49,x]+F[51,x]
F[49,x] = F[50,x]*F[8,x]
F[50,x] = F[15,x]+F[48,x]
F[51,x] = F[47,x]*F[8,x]
F[52,x] = F[14,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[55,x]*F[8,x]
F[55,x] = F[11,x]+F[25,x]
F[56,x] = F[57,x]+F[60,x]
F[57,x] = F[46,x]*F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[46,x]*F[8,x]
F[60,x] = F[18,x]^2*F[14,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_{41}\! \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_{24}\! \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_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{29}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right) F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{56}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{18}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{26}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right) F_{8}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right) F_{8}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right) F_{8}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = F_{46}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{46}\! \left(x \right) F_{8}\! \left(x \right)
F_{60}\! \left(x \right) = F_{18} \left(x \right)^{2} F_{14}\! \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_41(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_24(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))
Eq(F_22(x), F_23(x)*F_8(x))
Eq(F_23(x), F_15(x) + F_21(x))
Eq(F_24(x), F_18(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_32(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_28(x)*F_8(x))
Eq(F_28(x), F_15(x) + F_29(x))
Eq(F_29(x), F_26(x) + F_27(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_8(x))
Eq(F_31(x), F_18(x) + F_29(x))
Eq(F_32(x), F_33(x)*F_8(x))
Eq(F_33(x), F_31(x) + F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_8(x))
Eq(F_37(x), F_18(x) + F_35(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x)*F_8(x))
Eq(F_40(x), F_29(x) + F_38(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x)*F_8(x))
Eq(F_43(x), F_44(x) + F_56(x))
Eq(F_44(x), F_45(x) + F_52(x))
Eq(F_45(x), F_20(x)*F_46(x))
Eq(F_46(x), F_14(x) + F_47(x))
Eq(F_47(x), F_18(x) + F_48(x))
Eq(F_48(x), F_26(x) + F_49(x) + F_51(x))
Eq(F_49(x), F_50(x)*F_8(x))
Eq(F_50(x), F_15(x) + F_48(x))
Eq(F_51(x), F_47(x)*F_8(x))
Eq(F_52(x), F_14(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_55(x)*F_8(x))
Eq(F_55(x), F_11(x) + F_25(x))
Eq(F_56(x), F_57(x) + F_60(x))
Eq(F_57(x), F_46(x)*F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_46(x)*F_8(x))
Eq(F_60(x), F_14(x)*F_18(x)**2)
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": [1, 0, 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": [2, 3, 0, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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