0132_0213_0231_1230_3102
Counting sequence:
1, 1, 2, 6, 19, 54, 135, 307, 655, 1342, 2684, 5299, 10403, 20399, 40050, 78817, 155522, 307653, 609940, 1211452, 2409702, 4798713, 9564917, 19078576, 38076014, 76023094, 151840575, 303352242, 606176985, 1211506192, 2421647589, 4841095209, 9678640673, 19351549384, 38693837890, 77372707217, 154721213117, 309403288753, 618743275840, 1237384154887, 2474602659412, 4948937326063, 9897441070282, 19794180634746, 39587226258784, 79172616086859, 158342260827715, 316679713984502, 633351649069040, 1266690711696192, 2533361058192469, 5066689164898944, 10133325013282643, 20266563758750594, 40532987933374755, 81065750015029967, 162131134594454787, 324261677901846178, 648522399081306840, 1297043250153470847, 2594083995575743847, 5188163938411503363, 10376321319352208222, 20752632028494046141, 41505246889307366454, 83010466000724111849, 166020887055877385680, 332041701388293856712, 664083285107556542570, 1328166379822621620717, 2656332451583721268449, 5312664404713429805004, 10625328002911325655938, 21250654700853105424154, 42501307290221131854395, 85002611163987639726942, 170005216800035577432157, 340010424655676829871428, 680020834839019633798065, 1360041646261310917789957, 2720083254633559461024757, 5440166447961328198950988, 10880332796727803301580862, 21760665432954962785147261, 43521330606214428657439649, 87042660792232716585569189, 174085320904573795530614220, 348170640709059812693119747, 696341279638140209379420200, 1392682556396213224385817267, 2785365108132379838393791654, 5570730208724645872038815014, 11141460405249131328953218308, 22282920790757988438035561439, 44565841549575542241078249935, 89131683047470375627295278642, 178263366011319607764739106852, 356526731887337363184768306628, 713053463555751730534978145265, 1426106926757278612890690934092, 2852213852941409381767561195767
Generating function in Maple syntax:
(x^9+x^8-9*x^7+9*x^6-x^5-15*x^4+27*x^3-20*x^2+7*x-1)/(2*x-1)/(x^2+x-1)/(-1+x)^5
Generating function in latex syntax:
\frac{x^{9}+x^{8}-9 x^{7}+9 x^{6}-x^{5}-15 x^{4}+27 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(-1+x \right)^{5}}
Generating function in sympy syntax:
(x**9 + x**8 - 9*x**7 + 9*x**6 - x**5 - 15*x**4 + 27*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**5*(2*x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2+x-1)*(-1+x)^5*F(x)+x^9+x^8-9*x^7+9*x^6-x^5-15*x^4+27*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}+x -1\right) \left(-1+x \right)^{5} F \! \left(x \right)+x^{9}+x^{8}-9 x^{7}+9 x^{6}-x^{5}-15 x^{4}+27 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/120*(-84*5^(1/2)+180)*(-1/2*5^(1/2)-1/2)^(-n)+1/120*(84*5^(1/2)+180)*(1/2*5^(1/2)-1/2)^(-n)+1/24*n^4-1/4*n^3-37/24*n^2-1/4*n+9/4*2^n-7)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-84 \sqrt{5}+180\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{120}+\frac{\left(84 \sqrt{5}+180\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{4}\\-\frac{37 n^{2}}{24}-\frac{n}{4}+\frac{9 \,2^{n}}{4}-7 & \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) = 54
a(6) = 135
a(7) = 307
a(8) = 655
a(9) = 1342
a(n+3) = 1/24*n^4-7/12*n^3-13/24*n^2-2*a(n)-a(n+1)+3*a(n+2)+97/12*n-3, 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) = 54
a \! \left(6\right) = 135
a \! \left(7\right) = 307
a \! \left(8\right) = 655
a \! \left(9\right) = 1342
a \! \left(n +3\right) = \frac{n^{4}}{24}-\frac{7 n^{3}}{12}-\frac{13 n^{2}}{24}-2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{97 n}{12}-3, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18170/
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[14,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[7,x]
F[14,x] = F[15,x]+F[24,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[6,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[15,x]
F[23,x] = F[19,x]
F[24,x] = F[25,x]+F[26,x]+F[56,x]
F[25,x] = 0
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[38,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[25,x]+F[30,x]+F[35,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[11,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[32,x]*F[4,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[13,x]+F[37,x]
F[37,x] = F[29,x]
F[38,x] = F[24,x]+F[39,x]
F[39,x] = 2*F[25,x]+F[40,x]+F[49,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[11,x]+F[46,x]
F[46,x] = F[30,x]
F[47,x] = F[42,x]+F[48,x]
F[48,x] = F[40,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[19,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[23,x]+F[55,x]
F[55,x] = F[53,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[15,x]+F[42,x]
F[59,x] = F[60,x]
F[60,x] = F[25,x]+F[61,x]+F[68,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[28,x]+F[63,x]
F[63,x] = F[19,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[23,x]+F[67,x]
F[67,x] = F[64,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[22,x]+F[70,x]
F[70,x] = F[60,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_{14}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{6}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{56}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{38}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{29}\! \left(x \right)
F_{38}\! \left(x \right) = F_{24}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \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_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{30}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{40}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{23}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{25}\! \left(x \right)+F_{61}\! \left(x \right)+F_{68}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{28}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{19}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{23}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{64}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{22}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{60}\! \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_14(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_7(x))
Eq(F_14(x), F_15(x) + F_24(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_6(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_15(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_56(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_38(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_25(x) + F_30(x) + F_35(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_11(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_32(x)*F_4(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_13(x) + F_37(x))
Eq(F_37(x), F_29(x))
Eq(F_38(x), F_24(x) + F_39(x))
Eq(F_39(x), 2*F_25(x) + F_40(x) + F_49(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_11(x) + F_46(x))
Eq(F_46(x), F_30(x))
Eq(F_47(x), F_42(x) + F_48(x))
Eq(F_48(x), F_40(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_19(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_23(x) + F_55(x))
Eq(F_55(x), F_53(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_15(x) + F_42(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_25(x) + F_61(x) + F_68(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_28(x) + F_63(x))
Eq(F_63(x), F_19(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_23(x) + F_67(x))
Eq(F_67(x), F_64(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_22(x) + F_70(x))
Eq(F_70(x), F_60(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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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