0132_1023_1032_1230
Counting sequence:
1, 1, 2, 6, 20, 64, 195, 573, 1638, 4586, 12638, 34410, 92831, 248683, 662616, 1758282, 4650984, 12273020, 32326139, 85023729, 223385282, 586418362, 1538451642, 4034115366, 10574276535, 27709517999, 72595949060, 190161776058, 498056440628, 1304341940296, 3415638608403, 8943913051845, 23418780030846, 61317787868618, 160545308241206, 420339591058242, 1100516381221199, 2881295397932755, 7543541523865104, 19749672629623914, 51706163330947872, 135370191382505972, 354407158996564907, 927856782196004601, 2429174181139329050, 6359687748916817434, 16649933041970116050, 43590199331279913918, 114120840862980112743, 298772675083987516631, 782197888048280460668, 2047822396390199869530, 5361272115798404952236, 14035999580385330348928, 36746737884163754367171, 96204236589791950600077, 251866016920703351979414, 659393904243493514584490, 1726315875952440129271694, 4519554083899657767384474, 11832347096319012099353087, 30977488646203658540939707, 81100121724586662839481672, 212322882292149190062344586, 555868536681052228265524440, 1455282750809399198771275244, 3809979761863943439071789915, 9974656627016050985641569153, 26113990303651487640347759858, 68367314652873030293641785466, 178987954392836940458408030058, 468596550001376628131843845590, 1226801698562770881153597545751, 3211808551589892315492448500863, 8408623968012819354497249963636, 22014063376060393440923359623162, 57633566207392018157544502180004, 150886635340562978328254104492600, 395026340003191556141729894284659, 1034192385046800976365384523197909, 2707550815892789957852170511680302, 7088460064142726086986894410146442, 18557829379557702715061170187625062, 48585028080575010934463327488280946, 127197254874256587925584754018401455, 333006736566373268653886733517190179, 871822954873220249881073478971681280, 2282462128150001545038220118280756138, 5975563429770212513912056490180248848, 15644228161547492254994475715429048676, 40957121055645976769184707301538605323, 107227135006937863091246190533693510025, 280724283968262462585907107961292328266, 734945716904039224835621646700064526298, 1924112866756234612269671039837780516226, 5037392883389423412687678095236532960622, 13188065783461553227249216878742944811719, 34526804467094271472015820401784666343335, 90392347618019331594781361031270510502124, 236650238387359864124410057679470616253594, 619558367544852542402799384543228405508892
Generating function in Maple syntax:
-(2*x^7+2*x^5-4*x^4-6*x^3+12*x^2-6*x+1)/(x-1)/(2*x-1)/(x^2-3*x+1)/(x^2+x-1)
Generating function in latex syntax:
-\frac{2 x^{7}+2 x^{5}-4 x^{4}-6 x^{3}+12 x^{2}-6 x +1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right)}
Generating function in sympy syntax:
(-2*x**7 - 2*x**5 + 4*x**4 + 6*x**3 - 12*x**2 + 6*x - 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)*(x^2-3*x+1)*(x^2+x-1)*F(x)+2*x^7+2*x^5-4*x^4-6*x^3+12*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+2 x^{7}+2 x^{5}-4 x^{4}-6 x^{3}+12 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/10*(-9*5^(1/2)+30)*(3/2-1/2*5^(1/2))^(-n)+1/10*(9*5^(1/2)+30)*(3/2+1/2*5^(1/2))^(-n)+1/10*(-5*5^(1/2)+10)*(-1/2*5^(1/2)-1/2)^(-n)+1/10*(5*5^(1/2)+10)*(1/2*5^(1/2)-1/2)^(-n)-5*2^(-1+n)-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-9 \sqrt{5}+30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(9 \sqrt{5}+30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\\\frac{\left(-5 \sqrt{5}+10\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\frac{\left(5 \sqrt{5}+10\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}-5 \,2^{-1+n}-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) = 20
a(5) = 64
a(6) = 195
a(7) = 573
a(n+5) = -2*a(n)+5*a(n+1)+4*a(n+2)-11*a(n+3)+6*a(n+4)+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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 195
a \! \left(7\right) = 573
a \! \left(n +5\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)+4 a \! \left(n +2\right)-11 a \! \left(n +3\right)+6 a \! \left(n +4\right)+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15472/
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[35,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[16,x]+F[32,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[24,x]
F[22,x] = F[15,x]+F[16,x]+F[23,x]
F[23,x] = F[4,x]*F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[18,x]+F[30,x]
F[30,x] = F[27,x]+F[31,x]
F[31,x] = F[25,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[28,x]
F[35,x] = F[2,x]+F[36,x]
F[36,x] = F[15,x]+F[37,x]+F[59,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[22,x]+F[4,x]
F[40,x] = F[41,x]+F[43,x]
F[41,x] = F[15,x]+F[37,x]+F[42,x]
F[42,x] = F[2,x]*F[4,x]
F[43,x] = F[15,x]+F[44,x]+F[45,x]+F[58,x]
F[44,x] = 0
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[57,x]
F[47,x] = F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[27,x]+F[4,x]
F[52,x] = F[48,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[47,x]+F[56,x]
F[56,x] = F[53,x]
F[57,x] = F[45,x]
F[58,x] = F[36,x]*F[4,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[63,x]
F[61,x] = F[2,x]+F[62,x]
F[62,x] = F[49,x]
F[63,x] = F[36,x]+F[64,x]
F[64,x] = F[45,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_{35}\! \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_{16}\! \left(x \right)+F_{32}\! \left(x \right)
F_{15}\! \left(x \right) = 0
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_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{59}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)+F_{58}\! \left(x \right)
F_{44}\! \left(x \right) = 0
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_{57}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \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_{27}\! \left(x \right)+F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{47}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{53}\! \left(x \right)
F_{57}\! \left(x \right) = F_{45}\! \left(x \right)
F_{58}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{2}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{49}\! \left(x \right)
F_{63}\! \left(x \right) = F_{36}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{45}\! \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_35(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) + F_16(x) + F_32(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_24(x))
Eq(F_22(x), F_15(x) + F_16(x) + F_23(x))
Eq(F_23(x), F_4(x)*F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_18(x) + F_30(x))
Eq(F_30(x), F_27(x) + F_31(x))
Eq(F_31(x), F_25(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_28(x))
Eq(F_35(x), F_2(x) + F_36(x))
Eq(F_36(x), F_15(x) + F_37(x) + F_59(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_22(x) + F_4(x))
Eq(F_40(x), F_41(x) + F_43(x))
Eq(F_41(x), F_15(x) + F_37(x) + F_42(x))
Eq(F_42(x), F_2(x)*F_4(x))
Eq(F_43(x), F_15(x) + F_44(x) + F_45(x) + F_58(x))
Eq(F_44(x), 0)
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_57(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_49(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_27(x) + F_4(x))
Eq(F_52(x), F_48(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_47(x) + F_56(x))
Eq(F_56(x), F_53(x))
Eq(F_57(x), F_45(x))
Eq(F_58(x), F_36(x)*F_4(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_63(x))
Eq(F_61(x), F_2(x) + F_62(x))
Eq(F_62(x), F_49(x))
Eq(F_63(x), F_36(x) + F_64(x))
Eq(F_64(x), F_45(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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 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]]}], "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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 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]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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