0132_0213_0321_1320_2301_3021
Counting sequence:
1, 1, 2, 6, 18, 47, 114, 274, 671, 1682, 4294, 11091, 28838, 75246, 196683, 514542, 1346634, 3524999, 9227946, 24158362, 63246599, 165580826, 433495198, 1134904011, 2971215998, 7778743062, 20365012179, 53316292374, 139583863746, 365435297567, 956722027554, 2504730783586, 6557470321583, 17167680179426, 44945570214838, 117669030463107, 308061521172374, 806515533051774, 2111485077980571, 5527939700887422, 14472334024679034, 37889062373146871, 99194853094758618, 259695496911125866, 679891637638615703, 1779979416004717802, 4660046610375534094, 12200160415121880699, 31940434634990104046, 83621143489848427302, 218922995834555173539, 573147844013817088806, 1500520536206896088178, 3928413764606871170831, 10284720757613717419218, 26925748508234281081522, 70492524767089125819839, 184551825793033096372274, 483162952612010163291046, 1264937032042997393494707, 3311648143516982017186694, 8670007398507948658058766, 22698374052006863956982763, 59425114757512643212882446, 155576970220531065681657258, 407305795904080553832081767, 1066340417491710595814580234, 2791715456571051233611650874, 7308805952221443105020364071, 19134702400093278081449432762, 50095301248058391139327925374, 131151201344081895336534334251, 343358302784187294870275067998, 898923707008479989274290860086, 2353412818241252672952597502323, 6161314747715278029583501636662, 16130531424904581415797907397154, 42230279526998466217810220543999, 110560307156090817237632754223746, 289450641941273985495088042115842, 757791618667731139247631372112079, 1983924214061919432247806074208386, 5193981023518027157495786850500758, 13598018856492162040239554477281251, 35600075545958458963222876581330038, 93202207781383214849429075266695582, 244006547798191185585064349218743099, 638817435613190341905763972389519774, 1672445759041379840132227567949801946, 4378519841510949178490918731459871447, 11463113765491467695340528626429797434, 30010821454963453907530667147829505546, 78569350599398894027251472817058703543, 205697230343233228174223751303346589066, 538522340430300790495419781092981047278, 1409869790947669143312035591975596536027, 3691087032412706639440686994833808543694, 9663391306290450775010025392525829077574, 25299086886458645685589389182743678671171, 66233869353085486281758142155705206917702, 173402521172797813159685037284371942063314
Generating function in Maple syntax:
-(x^7-x^6-4*x^5+x^4-6*x^3+9*x^2-5*x+1)/(x^2-3*x+1)/(x-1)^3
Generating function in latex syntax:
-\frac{x^{7}-x^{6}-4 x^{5}+x^{4}-6 x^{3}+9 x^{2}-5 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**7 + x**6 + 4*x**5 - x**4 + 6*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**3*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^3*F(x)+x^7-x^6-4*x^5+x^4-6*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{7}-x^{6}-4 x^{5}+x^{4}-6 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/10*(5-5^(1/2))*(3/2-1/2*5^(1/2))^(-n)+1/10*(5+5^(1/2))*(3/2+1/2*5^(1/2))^(-n)+2*n^2-10*n+13)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(5-\sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(5+\sqrt{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+2 n^{2}-10 n +13 & \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) = 47
a(6) = 114
a(7) = 274
a(n+2) = -2*n^2-a(n)+3*a(n+1)+6*n-1, 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) = 18
a \! \left(5\right) = 47
a \! \left(6\right) = 114
a \! \left(7\right) = 274
a \! \left(n +2\right) = -2 n^{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+6 n -1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22241/
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[18,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[12,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[10,x]*F[4,x]
F[12,x] = F[13,x]+F[14,x]
F[13,x] = F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[4,x]
F[17,x] = F[15,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[54,x]
F[21,x] = F[22,x]+F[44,x]
F[22,x] = F[23,x]+F[27,x]
F[23,x] = F[24,x]*F[6,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[4,x]
F[27,x] = F[10,x]*F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[24,x]^2*F[30,x]*F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[35,x]
F[33,x] = F[1,x]+F[34,x]
F[34,x] = F[30,x]
F[35,x] = F[36,x]
F[36,x] = F[33,x]*F[37,x]*F[4,x]
F[37,x] = F[10,x]+F[38,x]
F[38,x] = F[25,x]+F[39,x]
F[39,x] = F[40,x]+F[41,x]+F[43,x]
F[40,x] = 0
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[13,x]+F[39,x]
F[43,x] = F[38,x]*F[4,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[51,x]
F[47,x] = F[10,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[33,x]*F[37,x]
F[50,x] = F[24,x]^2*F[4,x]
F[51,x] = F[24,x]^2*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[13,x]*F[4,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[64,x]
F[57,x] = F[10,x]*F[33,x]*F[58,x]
F[58,x] = F[10,x]+F[59,x]
F[59,x] = F[25,x]+F[60,x]
F[60,x] = F[40,x]+F[61,x]+F[63,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[13,x]+F[60,x]
F[63,x] = F[4,x]*F[59,x]
F[64,x] = F[24,x]^2*F[65,x]
F[65,x] = F[52,x]+F[66,x]
F[66,x] = F[10,x]*F[4,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_{18}\! \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_{12}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \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_{17}\! \left(x \right)+F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \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_{54}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{44}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24} \left(x \right)^{2} F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right) F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{10}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = 0
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{51}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{33}\! \left(x \right) F_{37}\! \left(x \right)
F_{50}\! \left(x \right) = F_{24} \left(x \right)^{2} F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{24} \left(x \right)^{2} F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{64}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right) F_{33}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{25}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{40}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{13}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{64}\! \left(x \right) = F_{24} \left(x \right)^{2} F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{52}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \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_18(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_12(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_10(x)*F_4(x))
Eq(F_12(x), F_13(x) + F_14(x))
Eq(F_13(x), F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_4(x))
Eq(F_17(x), F_15(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_54(x))
Eq(F_21(x), F_22(x) + F_44(x))
Eq(F_22(x), F_23(x) + F_27(x))
Eq(F_23(x), F_24(x)*F_6(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_4(x))
Eq(F_27(x), F_10(x)*F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_24(x)**2*F_30(x)*F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_35(x))
Eq(F_33(x), F_1(x) + F_34(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_33(x)*F_37(x)*F_4(x))
Eq(F_37(x), F_10(x) + F_38(x))
Eq(F_38(x), F_25(x) + F_39(x))
Eq(F_39(x), F_40(x) + F_41(x) + F_43(x))
Eq(F_40(x), 0)
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_13(x) + F_39(x))
Eq(F_43(x), F_38(x)*F_4(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_51(x))
Eq(F_47(x), F_10(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_33(x)*F_37(x))
Eq(F_50(x), F_24(x)**2*F_4(x))
Eq(F_51(x), F_24(x)**2*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_13(x)*F_4(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_64(x))
Eq(F_57(x), F_10(x)*F_33(x)*F_58(x))
Eq(F_58(x), F_10(x) + F_59(x))
Eq(F_59(x), F_25(x) + F_60(x))
Eq(F_60(x), F_40(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_13(x) + F_60(x))
Eq(F_63(x), F_4(x)*F_59(x))
Eq(F_64(x), F_24(x)**2*F_65(x))
Eq(F_65(x), F_52(x) + F_66(x))
Eq(F_66(x), F_10(x)*F_4(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": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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