0132_0213_1230_2103_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 120, 265, 562, 1165, 2383, 4838, 9780, 19720, 39698, 79824, 160367, 321945, 645924, 1295247, 2596142, 5201619, 10418595, 20862356, 41765824, 83598646, 167306270, 334789550, 669866307, 1340198255, 2681151008, 5363524053, 10729026810, 21461256825, 42927698343, 85863787714, 171741154652, 343504283460, 687044124634, 1374145785932, 2748384671527, 5496919985189, 10994083718540, 21988561834331, 43977761821654, 87956556201791, 175914783123995, 351832269536560, 703668913092552, 1407344903504370, 2814701258359558, 5629421045402218, 11258872070852315, 22517792650450555, 45035663789710888, 90071454576994529, 180143114640389762, 360286561764772325, 720573661499958719, 1441148193454346142, 2882297795333558212, 5764597869546435584, 11529199426397082210, 23058404818977722056, 46116819291443241759, 92233654202557869361, 184467333678274954420, 368934708249380544295, 737869482664750975422, 1475739072388322510635, 2951478318001455507283, 5902956916286542101708, 11805914286081525819920, 23611829305955124388910, 47223659799210763190958, 94447321519514113593894, 188894646147421328864947, 377789297324328346673415, 755578602786535484024304, 1511157218740435447728877, 3022314458786114165877322, 6044628952044836081918929, 12089257959867523184488855, 24178516009985505139862906, 48357032165999320071334764, 96714064568277408705239356, 193428129518861895764736170, 386856259656309638446381380, 773712520313512202164014167, 1547425042246503176516276925, 3094850087113378050492145308, 6189700178466606570632225523, 12379400363793435308372075750, 24758800738686943253499813015, 49517601495334181310863017803, 99035203019728730062345198184, 198070406086478122369173063832, 396140812249037274423448074730, 792281624621176240776480885014, 1584563249441535203167648577762, 3169126499205354819879568828107
Generating function in Maple syntax:
(4*x^9-10*x^8+9*x^7+2*x^4-13*x^3+14*x^2-6*x+1)/(x^2+x-1)/(2*x-1)/(x-1)^4
Generating function in latex syntax:
\frac{4 x^{9}-10 x^{8}+9 x^{7}+2 x^{4}-13 x^{3}+14 x^{2}-6 x +1}{\left(x^{2}+x -1\right) \left(2 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(4*x**9 - 10*x**8 + 9*x**7 + 2*x**4 - 13*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(2*x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(2*x-1)*(x-1)^4*F(x)+4*x^9-10*x^8+9*x^7+2*x^4-13*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{9}-10 x^{8}+9 x^{7}+2 x^{4}-13 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/30*(3*5^(1/2)-45)*(-1/2*5^(1/2)-1/2)^(-n)+1/30*(-3*5^(1/2)-45)*(1/2*5^(1/2)-1/2)^(-n)+1/6*n^3-3/2*n^2+7/3*n+5*2^(-1+n)-5)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(3 \sqrt{5}-45\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{30}+\frac{\left(-3 \sqrt{5}-45\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-\frac{3 n^{2}}{2}+\\\frac{7 n}{3}+5 \,2^{-1+n}-5 & \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) = 51
a(6) = 120
a(7) = 265
a(8) = 562
a(9) = 1165
a(n+3) = -2*a(n)-a(n+1)+3*a(n+2)+1/6*(-2+n)*(n^2-13*n+18), n >= 10
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) = 51
a \! \left(6\right) = 120
a \! \left(7\right) = 265
a \! \left(8\right) = 562
a \! \left(9\right) = 1165
a \! \left(n +3\right) = -2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{\left(-2+n \right) \left(n^{2}-13 n +18\right)}{6}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18482/
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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[27,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[6,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[18,x]
F[26,x] = F[22,x]
F[27,x] = F[28,x]+F[29,x]+F[48,x]
F[28,x] = 0
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[42,x]
F[31,x] = F[32,x]+F[7,x]
F[32,x] = F[28,x]+F[33,x]+F[39,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[11,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,x]*F[4,x]
F[38,x] = F[14,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[16,x]+F[41,x]
F[41,x] = F[32,x]
F[42,x] = F[22,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[22,x]
F[47,x] = F[43,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[58,x]
F[50,x] = F[18,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[11,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[35,x]
F[58,x] = F[59,x]+F[63,x]
F[59,x] = F[28,x]+F[29,x]+F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[25,x]+F[62,x]
F[62,x] = F[59,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = 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_{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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{48}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{42}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \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_{32}\! \left(x \right)
F_{42}\! \left(x \right) = F_{22}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{22}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{58}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \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_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{35}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{63}\! \left(x \right)
F_{59}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{25}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{59}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 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_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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_27(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_6(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_18(x))
Eq(F_26(x), F_22(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_48(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_42(x))
Eq(F_31(x), F_32(x) + F_7(x))
Eq(F_32(x), F_28(x) + F_33(x) + F_39(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_11(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(x)*F_4(x))
Eq(F_38(x), F_14(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_16(x) + F_41(x))
Eq(F_41(x), F_32(x))
Eq(F_42(x), F_22(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_22(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_58(x))
Eq(F_50(x), F_18(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_11(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_35(x))
Eq(F_58(x), F_59(x) + F_63(x))
Eq(F_59(x), F_28(x) + F_29(x) + F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_25(x) + F_62(x))
Eq(F_62(x), F_59(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_14(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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 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": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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