0132_1032_1203_1320
Counting sequence:
1, 1, 2, 6, 20, 65, 204, 622, 1854, 5429, 15676, 44759, 126646, 355706, 992984, 2757951, 7627352, 21017585, 57734784, 158168182, 432289938, 1179034985, 3209770612, 8723696171, 23674260970, 64159515890, 173662145324, 469518046707, 1268061378704, 3421386645509, 9222889515636, 24840514788334, 66850690137126, 179772616660061, 483094549569964, 1297324355565311, 3481674991818142, 9338243434453610, 25031818561067264, 67062817448836647, 179574394078407368, 480607439169769625, 1285670149337508072, 3437720069558550886, 9188013792898694970, 24546564728865018065, 65551864929085705444, 174989295164855914067, 466956538990507404754, 1245621422859278610914, 3321590127292047014996, 8854454259430033885659, 23596004536051734323456, 62860946396142060327341, 167414757154171999649244, 445739691732075847527646, 1186445467389866045599182, 3157163734394356639385285, 8399105541222306108604444, 22338765042912253793819687, 59398965756615669602968198, 157904847589806787876524890, 419673944914706945218830632, 1115145371389865908333307151, 2962488945700963201849246520, 7868473393810096909914566465, 20894660208986829605721290064, 55474542154497613147012015702, 147254341902327904905527030754, 390805575284068346379517747769, 1036988282906506163233954300756, 2751119877374716620724059892859, 7297427259643843884412780691386, 19353369023952421699782073550546, 51318245258939012255164262675324, 136055855950280576061760046267331, 360657224696104301152923653167472, 955885035169399116998448710022485, 2533097629631986640040620862528324, 6711737882810142030867087137060302, 17781006356530379306128754449890678, 47099622169476524275618951092719405, 124743992759382640009874565277425292, 330342412942608251326448966303648591, 874687283950982207456415897423253870, 2315721495700145510993796269570774090, 6130079335587978619106436811325071184, 16225320851491633095730123357756586487, 42940694107533660788299497757313313704, 113630389796220363508298397320934633449, 300656549367001334182989181124973247416, 795423852235650199026043683728081412934, 2104162715043348074549292536673285846666, 5565622822066992600635297133214960400801, 14719833630958409303310698634103242955204, 38926703181010431295271535164450021475491, 102931623362822985662706174161810472313762, 272149384149393980403904083228301274678210, 719488833360187437634746585971907899658404, 1901952811513257810351885948664283714685867, 5027304383650095013980278965535988426211808
Generating function in Maple syntax:
-(x^7-4*x^6+8*x^5-33*x^4+49*x^3-31*x^2+9*x-1)/(2*x-1)^2/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{x^{7}-4 x^{6}+8 x^{5}-33 x^{4}+49 x^{3}-31 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-x**7 + 4*x**6 - 8*x**5 + 33*x**4 - 49*x**3 + 31*x**2 - 9*x + 1)/((2*x - 1)**2*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x^2-3*x+1)^2*F(x)+x^7-4*x^6+8*x^5-33*x^4+49*x^3-31*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+x^{7}-4 x^{6}+8 x^{5}-33 x^{4}+49 x^{3}-31 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/200*((-20*n-56)*5^(1/2)+60*n+200)*(3/2-1/2*5^(1/2))^(-n)+1/200*((20*n+56)*5^(1/2)+60*n+200)*(3/2+1/2*5^(1/2))^(-n)-1/8*2^n*(n+2))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(\left(-20 n -56\right) \sqrt{5}+60 n +200\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{200}+\\\frac{\left(\left(20 n +56\right) \sqrt{5}+60 n +200\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{200}-\frac{2^{n} \left(n +2\right)}{8} & \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) = 65
a(6) = 204
a(7) = 622
a(n+6) = -4*a(n)+28*a(n+1)-69*a(n+2)+74*a(n+3)-39*a(n+4)+10*a(n+5), 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) = 65
a \! \left(6\right) = 204
a \! \left(7\right) = 622
a \! \left(n +6\right) = -4 a \! \left(n \right)+28 a \! \left(n +1\right)-69 a \! \left(n +2\right)+74 a \! \left(n +3\right)-39 a \! \left(n +4\right)+10 a \! \left(n +5\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15500/
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[21,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[6,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[10,x]
F[19,x] = F[14,x]
F[20,x] = F[12,x]*F[13,x]
F[21,x] = F[22,x]+F[6,x]
F[22,x] = F[15,x]+F[23,x]+F[36,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[29,x]
F[25,x] = F[10,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[10,x]+F[26,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[15,x]+F[23,x]+F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[30,x]+F[6,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[14,x]+F[33,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[32,x]+F[38,x]
F[38,x] = F[39,x]+F[49,x]
F[39,x] = F[15,x]+F[16,x]+F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[13,x]+F[42,x]
F[42,x] = F[39,x]+F[43,x]
F[43,x] = F[15,x]+F[44,x]+F[45,x]+F[48,x]
F[44,x] = 0
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[19,x]+F[47,x]
F[47,x] = F[43,x]
F[48,x] = F[12,x]*F[42,x]
F[49,x] = F[15,x]+F[50,x]+F[51,x]+F[55,x]
F[50,x] = 0
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[30,x]
F[54,x] = F[49,x]
F[55,x] = F[12,x]*F[38,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_{21}\! \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_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{36}\! \left(x \right)
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_{29}\! \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_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{40}\! \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_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = 0
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{19}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{50}\! \left(x \right) = 0
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{30}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \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_21(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_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_10(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_12(x)*F_13(x))
Eq(F_21(x), F_22(x) + F_6(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_36(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_29(x))
Eq(F_25(x), F_10(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_10(x) + F_26(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_15(x) + F_23(x) + F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_30(x) + F_6(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_14(x) + F_33(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_32(x) + F_38(x))
Eq(F_38(x), F_39(x) + F_49(x))
Eq(F_39(x), F_15(x) + F_16(x) + F_40(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_39(x) + F_43(x))
Eq(F_43(x), F_15(x) + F_44(x) + F_45(x) + F_48(x))
Eq(F_44(x), 0)
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_19(x) + F_47(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_12(x)*F_42(x))
Eq(F_49(x), F_15(x) + F_50(x) + F_51(x) + F_55(x))
Eq(F_50(x), 0)
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_30(x))
Eq(F_54(x), F_49(x))
Eq(F_55(x), F_12(x)*F_38(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, 3, 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": [1, 3, 2, 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, 3, 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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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