0213_1230_1302_2301_3021
Counting sequence:
1, 1, 2, 6, 19, 52, 128, 296, 657, 1416, 2986, 6196, 12707, 25844, 52260, 105264, 211477, 424144, 849758, 1701308, 3404775, 6812124, 13627288, 27258136, 54520409, 109045592, 218096658, 436199556, 872406187, 1744820356, 3489649676, 6979309376, 13958629917, 27917272224, 55834558150, 111669131404, 223338279407, 446676577004, 893353173888, 1786706369448, 3573412762465, 7146825550504, 14293651128698, 28587302287316, 57174604606899, 114349209248532, 228698418534388, 457396837108816, 914793674260517, 1829587348566896, 3659174697182766, 7318349394417756, 14636698788891127, 29273397577841404, 58546795155745640, 117093590311557944, 234187180623186537, 468374361246447864, 936748722492974818, 1873497444986033188, 3746994889972154555, 7493989779944402084, 14987979559888902108, 29975959119777907296, 59951918239555922989, 119903836479111959872, 239807672958224039318, 479615345916448204076, 959230691832896539647, 1918461383665793217036, 3836922767331586578256, 7673845534663173307336, 15347691069326346772337, 30695382138652693709384, 61390764277305387590730, 122781528554610775360884, 245563057109221550908867, 491126114218443102012724, 982252228436886204228548, 1964504456873772408668528, 3929008913747544817557045, 7858017827495089635342864, 15716035654990179270923518, 31432071309980358542094076, 62864142619960717084444679, 125728285239921434169155612, 251456570479842868338587448, 502913140959685736677461336, 1005826281919371473355219577, 2011652563838742946710746776, 4023305127677485893421812146, 8046610255354971786843954116, 16093220510709943573688249547, 32186441021419887147376852164, 64372882042839774294754069420, 128745764085679548589508516224, 257491528171359097179017422397, 514983056342718194358035247584, 1029966112685436388716070911078, 2059932225370872777432142251468, 4119864450741745554864284945935
Generating function in Maple syntax:
-(2*x^7+2*x^6-3*x^5+4*x^4-8*x^3+10*x^2-5*x+1)/(2*x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{2 x^{7}+2 x^{6}-3 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-2*x**7 - 2*x**6 + 3*x**5 - 4*x**4 + 8*x**3 - 10*x**2 + 5*x - 1)/((x - 1)**4*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^4*F(x)+2*x^7+2*x^6-3*x^5+4*x^4-8*x^3+10*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+2 x^{7}+2 x^{6}-3 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,13+13/4*2^n-31/2*n+3*n^2-1/2*n^3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 13+\frac{13 \,2^{n}}{4}-\frac{31 n}{2}+3 n^{2}-\frac{n^{3}}{2} & \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) = 52
a(6) = 128
a(7) = 296
a(n+1) = 2*a(n)+1/2*(-2+n)*(n^2-7*n+26), 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) = 52
a \! \left(6\right) = 128
a \! \left(7\right) = 296
a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(-2+n \right) \left(n^{2}-7 n +26\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20580/
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[16,x]*F[4,x]
F[4,x] = F[26,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[21,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[14,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[14,x]+F[25,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]+F[42,x]
F[27,x] = F[28,x]
F[28,x] = F[16,x]*F[29,x]
F[29,x] = F[13,x]+F[30,x]
F[30,x] = F[27,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[37,x]
F[32,x] = 0
F[33,x] = F[16,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[14,x]
F[36,x] = F[31,x]
F[37,x] = F[16,x]*F[38,x]
F[38,x] = F[14,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[16,x]*F[41,x]
F[41,x] = F[14,x]+F[39,x]
F[42,x] = F[32,x]+F[43,x]+F[60,x]
F[43,x] = F[16,x]*F[44,x]
F[44,x] = F[45,x]+F[49,x]
F[45,x] = F[46,x]+F[6,x]
F[46,x] = F[47,x]
F[47,x] = F[16,x]*F[48,x]
F[48,x] = F[6,x]
F[49,x] = F[42,x]+F[50,x]
F[50,x] = 2*F[32,x]+F[51,x]+F[55,x]
F[51,x] = F[16,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[46,x]
F[54,x] = F[50,x]
F[55,x] = F[16,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[32,x]+F[58,x]+F[60,x]
F[58,x] = F[16,x]*F[59,x]
F[59,x] = F[57,x]+F[6,x]
F[60,x] = F[16,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[14,x]+F[57,x]
F[63,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{26}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
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_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{42}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{16}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{32}\! \left(x \right)+F_{43}\! \left(x \right)+F_{60}\! \left(x \right)
F_{43}\! \left(x \right) = F_{16}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{6}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{6}\! \left(x \right)
F_{49}\! \left(x \right) = F_{42}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{51}\! \left(x \right) = F_{16}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{46}\! \left(x \right)
F_{54}\! \left(x \right) = F_{50}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{32}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{16}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{57}\! \left(x \right)+F_{6}\! \left(x \right)
F_{60}\! \left(x \right) = F_{16}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{14}\! \left(x \right)+F_{57}\! \left(x \right)
F_{63}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_26(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_16(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_21(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_14(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_14(x) + F_25(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x) + F_42(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_16(x)*F_29(x))
Eq(F_29(x), F_13(x) + F_30(x))
Eq(F_30(x), F_27(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_37(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_16(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_14(x))
Eq(F_36(x), F_31(x))
Eq(F_37(x), F_16(x)*F_38(x))
Eq(F_38(x), F_14(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_16(x)*F_41(x))
Eq(F_41(x), F_14(x) + F_39(x))
Eq(F_42(x), F_32(x) + F_43(x) + F_60(x))
Eq(F_43(x), F_16(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_49(x))
Eq(F_45(x), F_46(x) + F_6(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_16(x)*F_48(x))
Eq(F_48(x), F_6(x))
Eq(F_49(x), F_42(x) + F_50(x))
Eq(F_50(x), 2*F_32(x) + F_51(x) + F_55(x))
Eq(F_51(x), F_16(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_46(x))
Eq(F_54(x), F_50(x))
Eq(F_55(x), F_16(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_32(x) + F_58(x) + F_60(x))
Eq(F_58(x), F_16(x)*F_59(x))
Eq(F_59(x), F_57(x) + F_6(x))
Eq(F_60(x), F_16(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_14(x) + F_57(x))
Eq(F_63(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 0, 2, 1], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 0, 2, 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