0132_1230_1302_1320_2130
Counting sequence:
1, 1, 2, 6, 19, 58, 171, 489, 1366, 3747, 10132, 27084, 71721, 188444, 491869, 1276623, 3297260, 8479877, 21726630, 55481106, 141254399, 358668990, 908516111, 2296227765, 5791948482, 14582604967, 36653095736, 91983151320, 230504045461, 576854891296, 1441825773249, 3599593977243, 8976800751832, 22363857068553, 55661625088330, 138411989231646, 343892477723883, 853735575010370, 2117847762578803, 5249938000167297, 13005221697746222, 32195884165324715, 79655493502558428, 196959380888432100, 486737778189566465, 1202214492805843236, 2967889397787953189, 7323218111893428327, 18061497902584867524, 44525783302594957453, 109719375559848428270, 270256725911970899754, 665423521415222045143, 1637777348294957710534, 4029516071141654724567, 9910558776403844396877, 24366670048737015251098, 59889721009278795601839, 147153792762884119593088, 361458490061626981767408, 887600820634887490225005, 2178981410355480723391016, 5347736247142755866451337, 13121120395258885076355891, 32185482624693218188731824, 78929763309328598413019153, 193515870159749661102738066, 474340903126309689753630774, 1162427336323731824968238531, 2848034295093980677541721162, 6976403025063470640113146427, 17085493261644684215742558489, 41834602479752141047608817926, 102413777000371332520955847859, 250667526970338840485522375204, 613418650699959448493999974908, 1500849838377922641873522939865, 3671478167230044976043046460972, 8979870862394157985963617689325, 21959709110904891975780286100991, 53692332211533149385148200241564, 131258949007516136669134711546005, 320832424300984522017157683607606, 784082761231868325213988224271170, 1915938068083906560759950483443887, 4680998103660435367874060039256782, 11434952809245470527103228609447071, 27929979996093516804411004201228005, 68210083883157478131181368945546866, 166559566199800561439616492902686391, 406663224239267751751355988305292744, 992763449028746454797054487432382728, 2423278998954090591795858632562652997, 5914376634601997889144285109261989168, 14433231520145556802045849233887827025, 35218293366533122607914337699344335051, 85925924888479294679192653259991076488, 209619810105666708403614255596462338585, 511320985659430197022368515834602034138, 1247122329505937069957560601406174818382, 3041442181393741757491855698309654774427
Generating function in Maple syntax:
(x^5-4*x^3-3*x^2+4*x-1)/(x-1)/(x^2+2*x-1)^2
Generating function in latex syntax:
\frac{x^{5}-4 x^{3}-3 x^{2}+4 x -1}{\left(x -1\right) \left(x^{2}+2 x -1\right)^{2}}
Generating function in sympy syntax:
(x**5 - 4*x**3 - 3*x**2 + 4*x - 1)/((x - 1)*(x**2 + 2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+2*x-1)^2*F(x)-x^5+4*x^3+3*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+2 x -1\right)^{2} F \! \left(x \right)-x^{5}+4 x^{3}+3 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/16*(2^(1/2)*n-3*2^(1/2)+4*n-6)*(-1-2^(1/2))^(-n)+3/4-1/16*(2^(1/2)*n-3*2^(1/2)-4*n+6)*(2^(1/2)-1)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\sqrt{2}\, n -3 \sqrt{2}+4 n -6\right) \left(-1-\sqrt{2}\right)^{-n}}{16}+\frac{3}{4}-\\\frac{\left(\sqrt{2}\, n -3 \sqrt{2}-4 n +6\right) \left(\sqrt{2}-1\right)^{-n}}{16} & \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) = 58
a(n+4) = -a(n)-4*a(n+1)-2*a(n+2)+4*a(n+3)+3, n >= 6
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) = 58
a \! \left(n +4\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-2 a \! \left(n +2\right)+4 a \! \left(n +3\right)+3, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/20036/
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[12,x]*F[4,x]
F[4,x] = F[32,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]+F[23,x]+F[31,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[10,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]*F[12,x]
F[28,x] = F[21,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[21,x]
F[31,x] = F[12,x]*F[13,x]
F[32,x] = F[14,x]+F[33,x]
F[33,x] = F[22,x]+F[34,x]+F[40,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[37,x]+F[6,x]
F[37,x] = F[27,x]
F[38,x] = F[33,x]+F[39,x]
F[39,x] = F[30,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[13,x]+F[42,x]
F[42,x] = F[43,x]+F[53,x]
F[43,x] = F[22,x]+F[44,x]+F[48,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[12,x]
F[47,x] = F[19,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[18,x]+F[50,x]
F[50,x] = F[43,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[43,x]
F[53,x] = F[22,x]+F[54,x]+F[58,x]+F[63,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[26,x]
F[57,x] = F[29,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[28,x]+F[60,x]
F[60,x] = F[53,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[12,x]*F[53,x]
F[63,x] = F[12,x]*F[42,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{32}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{21}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{28}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{22}\! \left(x \right)+F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{6}\! \left(x \right)
F_{37}\! \left(x \right) = F_{27}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{30}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{53}\! \left(x \right)
F_{43}\! \left(x \right) = F_{22}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \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_{12}\! \left(x \right)
F_{47}\! \left(x \right) = F_{19}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{43}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)+F_{63}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{26}\! \left(x \right)
F_{57}\! \left(x \right) = F_{29}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{28}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{53}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_32(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_31(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_10(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x)*F_12(x))
Eq(F_28(x), F_21(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_21(x))
Eq(F_31(x), F_12(x)*F_13(x))
Eq(F_32(x), F_14(x) + F_33(x))
Eq(F_33(x), F_22(x) + F_34(x) + F_40(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_37(x) + F_6(x))
Eq(F_37(x), F_27(x))
Eq(F_38(x), F_33(x) + F_39(x))
Eq(F_39(x), F_30(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_13(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_53(x))
Eq(F_43(x), F_22(x) + F_44(x) + F_48(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_12(x))
Eq(F_47(x), F_19(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_18(x) + F_50(x))
Eq(F_50(x), F_43(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_43(x))
Eq(F_53(x), F_22(x) + F_54(x) + F_58(x) + F_63(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_26(x))
Eq(F_57(x), F_29(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_28(x) + F_60(x))
Eq(F_60(x), F_53(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_12(x)*F_53(x))
Eq(F_63(x), F_12(x)*F_42(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": [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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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": [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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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