0231_0312_1203_2130_2301
Counting sequence:
1, 1, 2, 6, 19, 53, 136, 338, 839, 2107, 5364, 13806, 35797, 93231, 243424, 636428, 1665081, 4357843, 11407254, 29862472, 78178429, 204670761, 535831442, 1402820756, 3672627579, 9615058253, 25172542926, 65902565698, 172535148719, 451702874337, 1182573467444, 3096017520366, 8105479085187, 21220419725831, 55555780081984, 145446920508778, 380784981431921, 996908023773403, 2609939089873484, 6832909245830952, 17888788647601909, 46833456696955871, 122611581443245282, 321001287632757956, 840392281455004889, 2200175556732231253, 5760134388741661566, 15080227609492724208, 39480548439736479799, 103361417709716681817, 270603704689413530074, 708449696358523870526, 1854745384386158041227, 4855786456799950210381, 12712613986013692544544, 33282055501241127375178, 87133552517709689530111, 228118602051887941161363, 597222253637954133897164, 1563548158861974460470182, 4093422222947969247450189, 10716718509981933281813831, 28056733306997830597921272, 73453481411011558511876356, 192303710926036844937630449, 503457651367098976300933803, 1318069243175260083965085806, 3450750078158681275594234368, 9034180991300783742817523829, 23651792895743669952858239297, 61921197695930226115757091754, 162111800192047008394412929036, 424414202880210799067481583667, 1111130808448585388808031705381, 2908978222465545367356613410854, 7615803858948050713261808400378, 19938433354378606772428811658151, 52199496204187769604024626436473, 136660055258184702039645067508044, 357780669570366336514910575938662, 936681953452914307505086660153019, 2452265190788376586000349404359391, 6420113618912215450495961552757912, 16808075665948269765487535253740706, 44004113378932593845966644208284009, 115204264470849511772412397370924403, 301608680033615941471270547904295396, 789621775629998312641399246341760928, 2067256646856378996452927191120779309, 5412148164939138676717382327020361527, 14169187847961037033699219789940082234, 37095415378943972424380277042799654396, 97117058288870880239441611338458642257, 254255759487668668293944556972576025581, 665650220174135124642392059579269179414, 1742694901034736705633231621765231249128, 4562434482930074992257302805716424295791, 11944608547755488271138676795384041357233, 31271391160336389821158727580435699485874, 81869564933253681192337505945923056801142, 214337303639424653755853790257333470608899
Generating function in Maple syntax:
(2*x^3-6*x^2+4*x-1)*(x^3+2*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{\left(2 x^{3}-6 x^{2}+4 x -1\right) \left(x^{3}+2 x -1\right)}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**3 + 2*x - 1)*(2*x**3 - 6*x**2 + 4*x - 1)/((x - 1)**4*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^4*F(x)-(2*x^3-6*x^2+4*x-1)*(x^3+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)-\left(2 x^{3}-6 x^{2}+4 x -1\right) \left(x^{3}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(18*5^(1/2)-30)*(3/2-1/2*5^(1/2))^(-n)+1/30*(-18*5^(1/2)-30)*(3/2+1/2*5^(1/2))^(-n)+1/3*n^3-3/2*n^2+7/6*n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(18 \sqrt{5}-30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-18 \sqrt{5}-30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{3}-\frac{3 n^{2}}{2}+\\\frac{7 n}{6}+1 & \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) = 53
a(6) = 136
a(n+2) = -1/3*n^3+1/2*n^2-a(n)+3*a(n+1)+17/6*n-2, 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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 136
a \! \left(n +2\right) = -\frac{n^{3}}{3}+\frac{n^{2}}{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{17 n}{6}-2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20696/
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[13,x]*F[4,x]
F[4,x] = F[17,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = F[14,x]+F[6,x]
F[17,x] = F[18,x]+F[36,x]
F[18,x] = F[19,x]
F[19,x] = F[13,x]*F[20,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[13,x]*F[24,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[10,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[30,x]+F[32,x]
F[30,x] = F[10,x]+F[31,x]
F[31,x] = F[28,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[13,x]*F[35,x]
F[35,x] = F[10,x]+F[33,x]
F[36,x] = F[37,x]+F[38,x]+F[55,x]
F[37,x] = 0
F[38,x] = F[13,x]*F[39,x]
F[39,x] = F[40,x]+F[46,x]
F[40,x] = F[41,x]+F[6,x]
F[41,x] = F[42,x]
F[42,x] = F[13,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[10,x]
F[45,x] = F[33,x]
F[46,x] = F[36,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[13,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[33,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[13,x]*F[54,x]
F[54,x] = F[33,x]+F[52,x]
F[55,x] = F[13,x]*F[56,x]
F[56,x] = F[57,x]+F[61,x]
F[57,x] = F[10,x]+F[58,x]
F[58,x] = F[37,x]+F[55,x]+F[59,x]
F[59,x] = F[13,x]*F[60,x]
F[60,x] = F[58,x]+F[6,x]
F[61,x] = F[62,x]+F[67,x]
F[62,x] = F[37,x]+F[63,x]+F[65,x]
F[63,x] = F[13,x]*F[64,x]
F[64,x] = F[10,x]+F[33,x]
F[65,x] = F[13,x]*F[66,x]
F[66,x] = F[10,x]+F[62,x]
F[67,x] = F[68,x]
F[68,x] = F[13,x]*F[69,x]
F[69,x] = F[58,x]+F[67,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{17}\! \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_{13}\! \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_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
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) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{55}\! \left(x \right)
F_{37}\! \left(x \right) = 0
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{46}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{6}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)
F_{45}\! \left(x \right) = F_{33}\! \left(x \right)
F_{46}\! \left(x \right) = F_{36}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{33}\! \left(x \right)
F_{51}\! \left(x \right) = 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_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{33}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{37}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{13}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{58}\! \left(x \right)+F_{6}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{67}\! \left(x \right)
F_{62}\! \left(x \right) = F_{37}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{13}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{10}\! \left(x \right)+F_{33}\! \left(x \right)
F_{65}\! \left(x \right) = F_{13}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{10}\! \left(x \right)+F_{62}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{13}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{58}\! \left(x \right)+F_{67}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_17(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_14(x) + F_6(x))
Eq(F_17(x), F_18(x) + F_36(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_13(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_26(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_13(x)*F_24(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_10(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_32(x))
Eq(F_30(x), F_10(x) + F_31(x))
Eq(F_31(x), F_28(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_13(x)*F_35(x))
Eq(F_35(x), F_10(x) + F_33(x))
Eq(F_36(x), F_37(x) + F_38(x) + F_55(x))
Eq(F_37(x), 0)
Eq(F_38(x), F_13(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_46(x))
Eq(F_40(x), F_41(x) + F_6(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_13(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_10(x))
Eq(F_45(x), F_33(x))
Eq(F_46(x), F_36(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_13(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_33(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_13(x)*F_54(x))
Eq(F_54(x), F_33(x) + F_52(x))
Eq(F_55(x), F_13(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_61(x))
Eq(F_57(x), F_10(x) + F_58(x))
Eq(F_58(x), F_37(x) + F_55(x) + F_59(x))
Eq(F_59(x), F_13(x)*F_60(x))
Eq(F_60(x), F_58(x) + F_6(x))
Eq(F_61(x), F_62(x) + F_67(x))
Eq(F_62(x), F_37(x) + F_63(x) + F_65(x))
Eq(F_63(x), F_13(x)*F_64(x))
Eq(F_64(x), F_10(x) + F_33(x))
Eq(F_65(x), F_13(x)*F_66(x))
Eq(F_66(x), F_10(x) + F_62(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_13(x)*F_69(x))
Eq(F_69(x), F_58(x) + F_67(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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