0132_0213_1032_1203_1230
Counting sequence:
1, 1, 2, 6, 19, 57, 164, 458, 1254, 3388, 9071, 24135, 63934, 168834, 444848, 1170182, 3074499, 8070685, 21171920, 55513194, 145502914, 381264456, 998825687, 2616278891, 6852136794, 17944372162, 46989443324, 123040856418, 322166876779, 843527200593, 2208549456956, 5782390437770, 15139160073534, 39636165702964, 103771488043583, 271682599097967, 711284908412374, 1862189320938690, 4875317438299880, 12763831752525854, 33416315321477139, 87485389192146661, 229040402176356872, 599636917116466026, 1569872548629791194, 4109985127520827392, 10760091631160616359, 28170307359983378003, 73750865636132821554, 193082359921966791298, 505496354875034649716, 1323406986190700137530, 3464725166667384195259, 9070769639744310570969, 23747586004418677494068, 62171992877197616853002, 162768401634513011668374, 426133230040965779067820, 1115631324517546779795215, 2920760815569859884975639, 7646651266308177673013614, 20019193271586597294532866, 52410929124914871244790432, 137213595256083573786223286, 359229858949185416797893219, 940475986203169305244701997, 2462198108883711703471166144, 6446118358894737656768383338, 16876157004694034359823299954, 44182352728974414440999971128, 115670901329803279221883487351, 302830351555583518796494025915, 792820153927243395587825412234, 2075630111406738787138405393474, 5434070182654157013777746424428, 14226580441278100042034024022162, 37245671150624878189548694047499, 97510433029486003874545261881825, 255285627975612070824983954192300, 668346450972908081270715247905674, 1749753725094227914911324461039214, 4580914724612007141783643777356196, 11992990449346256458135983831526367, 31398056624635688113544727613540671, 82201179426978659620922310453263494, 215205481661135994188181364261441602, 563415265566100729760234312639833304, 1475040315056509008626551781180846798, 3861705679642111923026980802131916979, 10110076723947198014009814670762201333, 26468524492354224625693115160521791736, 69295496753424960875770943073898771050, 181417965768539628025922450808999613834, 474958400553431863248821903432745541104, 1243457235894231841811314180454157134663, 3255413307134215422362002433249190582947, 8522782685518317945620916596127597707874, 22312934749440545455180994148386846851330, 58416021562842932501262819161319563244260, 152935129939167480211257029525510093577834, 400389368254817964457755721086232360558939
Generating function in Maple syntax:
-(x^6+2*x^5-5*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{x^{6}+2 x^{5}-5 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:
(-x**6 - 2*x**5 + 5*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)+x^6+2*x^5-5*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)+x^{6}+2 x^{5}-5 x^{4}-6 x^{3}+12 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,17*(3/2-1/2*5^(1/2))^(-n+1)+17*(3/2+1/2*5^(1/2))^(-n+1)+17*(-1/2*5^(1/2)-1/2)^(-n+1)+17*(1/2*5^(1/2)-1/2)^(-n+1)+1/20*(169*5^(1/2)-495)*(3/2-1/2*5^(1/2))^(-n)+1/20*(-169*5^(1/2)-495)*(3/2+1/2*5^(1/2))^(-n)+1/20*(173*5^(1/2)+165)*(-1/2*5^(1/2)-1/2)^(-n)+1/20*(-173*5^(1/2)+165)*(1/2*5^(1/2)-1/2)^(-n)-2^(n-1)+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 17 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n +1}+17 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n +1}+17 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n +1}+\\17 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n +1}+\frac{\left(169 \sqrt{5}-495\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(-169 \sqrt{5}-495\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(173 \sqrt{5}+165\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\\\frac{\left(-173 \sqrt{5}+165\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}-2^{n -1}+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) = 57
a(6) = 164
a(n+5) = -2*a(n)+5*a(n+1)+4*a(n+2)-11*a(n+3)+6*a(n+4)-1, 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) = 57
a \! \left(6\right) = 164
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 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18322/
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[23,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[22,x] = F[12,x]*F[13,x]
F[23,x] = F[24,x]+F[44,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[24,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[43,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[12,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[28,x]
F[38,x] = F[39,x]+F[41,x]
F[39,x] = F[32,x]+F[33,x]+F[40,x]
F[40,x] = F[12,x]*F[24,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[31,x]
F[43,x] = F[12,x]*F[30,x]
F[44,x] = F[32,x]+F[45,x]+F[58,x]
F[45,x] = F[12,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[49,x] = F[12,x]*F[50,x]
F[50,x] = F[28,x]+F[48,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[32,x]+F[45,x]+F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[24,x]+F[52,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[31,x]+F[55,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[54,x]+F[60,x]
F[60,x] = F[61,x]+F[68,x]
F[61,x] = F[32,x]+F[33,x]+F[62,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[24,x]+F[39,x]
F[65,x] = F[61,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[12,x]*F[61,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]*F[60,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_{23}\! \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_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{44}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{43}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{44}\! \left(x \right) = F_{32}\! \left(x \right)+F_{45}\! \left(x \right)+F_{58}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \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_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{28}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{32}\! \left(x \right)+F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{24}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{31}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{68}\! \left(x \right)
F_{61}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{24}\! \left(x \right)+F_{39}\! \left(x \right)
F_{65}\! \left(x \right) = F_{61}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \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_23(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))
Eq(F_22(x), F_12(x)*F_13(x))
Eq(F_23(x), F_24(x) + F_44(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_24(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_43(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_12(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_28(x))
Eq(F_38(x), F_39(x) + F_41(x))
Eq(F_39(x), F_32(x) + F_33(x) + F_40(x))
Eq(F_40(x), F_12(x)*F_24(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_31(x))
Eq(F_43(x), F_12(x)*F_30(x))
Eq(F_44(x), F_32(x) + F_45(x) + F_58(x))
Eq(F_45(x), F_12(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))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_28(x) + F_48(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_32(x) + F_45(x) + F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_24(x) + F_52(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_31(x) + F_55(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_54(x) + F_60(x))
Eq(F_60(x), F_61(x) + F_68(x))
Eq(F_61(x), F_32(x) + F_33(x) + F_62(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_24(x) + F_39(x))
Eq(F_65(x), F_61(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_12(x)*F_61(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_12(x)*F_60(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:
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