0123_0132_0213_1203_1230_2013
Counting sequence:
1, 1, 2, 6, 18, 56, 178, 579, 1923, 6506, 22365, 77933, 274718, 977979, 3511113, 12698137, 46218040, 169171277, 622312677, 2299457564, 8530613261, 31761811465, 118646747526, 444535638601, 1670121406171, 6290484218924, 23748150402599, 89848292734623, 340609688743278, 1293633096297261, 4921733383632123, 18755518085853557, 71581343342349270, 273583719761992119, 1047040975499975757, 4012237929344782918, 15393189206405750639, 59123660470557673397, 227331115692249218452, 874978539243539207511, 3370968971638267131983, 12999015359059752872734, 50170160158049479746317, 193794817538918788011711, 749176237466156072927588, 2898381950672152230871619, 11221286660485488307669447, 43474206660194191278463116, 168542734139782698284031823, 653831638465211933844244531, 2537980233876816554796067010, 9857493052497110597761910553, 38308116916634231795452846163, 148953910083885143873306970284, 579483411136927595216084490847, 2255536194918066741580763645343, 8783557097843960891451287581654, 34221216916319241795292312387717, 133388320447595759725142601852827, 520152626777942283442587915782328, 2029214991243413477787506659049891, 7919605586351057440255888535314331, 30920856391447857658381308182559742, 120772056619401183585598419969821526, 471892607400574447071565853773381594, 1844491536053351339953783398094308220, 7212107388262605203061246872289741014, 28209486799970634668628603095953268484, 110375155018072900127887659220245827798, 432002694284346475426582950543143821082, 1691362109724781074592041505626216976480, 6623963723634801158323997063254309264512, 25949372006228417554865152782754708177492, 101685685245648385435559507090368154893804, 398578154783561324402102954255422715047136, 1562732515729433765984006708089054462962960, 6128725844254092359439423287701723370980136, 24041809777932653076977269830862398765611096, 94334961372992596073450072556421722453877792, 370240566872570379454288465858599798097381038, 1453444618969834330539991111710394318740201570, 5707075073480195393548953183016145765808452888, 22414385552172134014960819058632860807609660858, 88051321658455883151471341681305387349404010366, 345969988218789581525604342993275503304892483104, 1359666130533958911234336238648720055091429485790, 5344601290230482991217351483234565883983730722334, 21012874849585321027574759366559219164996543794024, 82630576578465668920259356679598151594804307148718, 324996980062720932877649266903473245559613042179462, 1278495636270015554986314599147320898591515368756980, 5030356533308739580233910557256460359752065346907042, 19795940790469934978489753619109040761632589663598022, 77916549391951080939350520611668450993806320557895464, 306731120055514242272499396926697528297691875454051886, 1207699714354212699561530660831646278344789496431975400, 4755887347779527317060069045890637020339426414288262386, 18731568822308459037933265812469436866722709345150184786, 73787927830877435366053294383644436901663853989437669572, 290712468735156093191772344898488662201002336034329391498, 1145533627516962309680620308140148018855462248312999748390
Generating function in Maple syntax:
1/2*(-2*(-4*x+1)^(1/2)*x^2+4*x^2+(-4*x+1)^(1/2)-1)/(x^2+x-1)/x
Generating function in latex syntax:
\frac{-2 \sqrt{-4 x +1}\, x^{2}+4 x^{2}+\sqrt{-4 x +1}-1}{2 \left(x^{2}+x -1\right) x}
Generating function in sympy syntax:
(-x**2*sqrt(1 - 4*x) + 2*x**2 + sqrt(1 - 4*x)/2 - 1/2)/(x*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x^2+x-1)^2*F(x)^2-(2*x+1)*(2*x-1)*(x^2+x-1)*F(x)+4*x^4+3*x^3-4*x^2-x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{2}+x -1\right)^{2} F \! \left(x \right)^{2}-\left(2 x +1\right) \left(2 x -1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+4 x^{4}+3 x^{3}-4 x^{2}-x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(n+5) = 4*(1+2*n)/(n+6)*a(n)+6*n/(n+6)*a(1+n)-2*(13+7*n)/(n+6)*a(n+2)-(8+n)/(n+6)*a(n+3)+(24+5*n)/(n+6)*a(n+4), n >= 5
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) = 18
a \! \left(n +5\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +6}+\frac{6 n a \! \left(1+n \right)}{n +6}-\frac{2 \left(13+7 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(8+n \right) a \! \left(n +3\right)}{n +6}+\frac{\left(24+5 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 5
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22062/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[46,x]
F[4,x] = F[5,x,1]
F[5,x,k[0]] = F[6,x,k[0]]
F[6,x,k[0]] = F[22,x,k[0]]*F[7,x,k[0]]
F[7,x,k[0]] = F[44,x,k[0]]+F[8,x,k[0]]
F[8,x,k[0]] = F[18,x,k[0]]+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[14,x]
F[12,x] = F[13,x]+F[15,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = x
F[15,x] = F[10,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[14,x]
F[18,x,k[0]] = F[19,x,k[0]]+F[23,x,k[0]]
F[19,x,k[0]] = F[20,x,k[0]]
F[20,x,k[0]] = F[21,x,k[0]]*F[22,x,k[0]]
F[21,x,k[0]] = F[1,x]+F[19,x,k[0]]
F[22,x,k[0]] = k[0]*x
F[23,x,k[0]] = F[24,x]+F[25,x,k[0]]+F[27,x,k[0]]
F[24,x] = 0
F[25,x,k[0]] = F[22,x,k[0]]*F[26,x,k[0]]
F[26,x,k[0]] = F[10,x]+F[23,x,k[0]]
F[27,x,k[0]] = F[14,x]*F[28,x,k[0]]
F[28,x,k[0]] = F[29,x,k[0]]+F[32,x,k[0]]
F[29,x,k[0]] = F[19,x,k[0]]+F[30,x,k[0]]
F[30,x,k[0]] = F[31,x,k[0]]
F[31,x,k[0]] = F[14,x]*F[19,x,k[0]]
F[32,x,k[0]] = F[33,x,k[0]]+F[42,x,k[0]]
F[33,x,k[0]] = F[24,x]+F[27,x,k[0]]+F[34,x,k[0]]
F[34,x,k[0]] = F[22,x,k[0]]*F[35,x,k[0]]
F[35,x,k[0]] = F[36,x]+F[39,x,k[0]]
F[36,x] = F[37,x]
F[37,x] = F[14,x]*F[38,x]
F[38,x] = F[1,x]+F[36,x]
F[39,x,k[0]] = F[24,x]+F[34,x,k[0]]+F[40,x,k[0]]
F[40,x,k[0]] = F[14,x]*F[41,x,k[0]]
F[41,x,k[0]] = F[19,x,k[0]]+F[39,x,k[0]]
F[42,x,k[0]] = F[43,x,k[0]]
F[43,x,k[0]] = F[14,x]*F[33,x,k[0]]
F[44,x,k[0]] = F[45,x,k[0]]
F[45,x,k[0]] = F[14,x]*F[21,x,k[0]]*F[9,x]
F[46,x] = F[14,x]*F[47,x]
F[47,x] = F[48,x,1]
F[48,x,k[0]] = F[1,x]+F[49,x,k[0]]+F[50,x,k[0]]+F[52,x,k[0]]
F[49,x,k[0]] = F[22,x,k[0]]*F[48,x,k[0]]
F[50,x,k[0]] = F[14,x]*F[51,x,k[0]]
F[51,x,k[0]] = (F[48,x,k[0]]*k[0]-F[48,x,1])/(k[0]-1)
F[52,x,k[0]] = F[53,x,k[0]]
F[53,x,k[0]] = F[14,x]*F[54,x,k[0]]
F[54,x,k[0]] = (F[7,x,k[0]]*k[0]-F[7,x,1])/(k[0]-1)
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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{46}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x , 1\right)
F_{5}\! \left(x , y\right) = F_{6}\! \left(x , y\right)
F_{6}\! \left(x , y\right) = F_{22}\! \left(x , y\right) F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{44}\! \left(x , y\right)+F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{18}\! \left(x , y\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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{15}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{10}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)+F_{23}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{20}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right) F_{22}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = y x
F_{23}\! \left(x , y\right) = F_{24}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{27}\! \left(x , y\right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x , y\right) = F_{22}\! \left(x , y\right) F_{26}\! \left(x , y\right)
F_{26}\! \left(x , y\right) = F_{10}\! \left(x \right)+F_{23}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{28}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = F_{29}\! \left(x , y\right)+F_{32}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{19}\! \left(x , y\right)+F_{30}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{31}\! \left(x , y\right)
F_{31}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{19}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{33}\! \left(x , y\right)+F_{42}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{24}\! \left(x \right)+F_{27}\! \left(x , y\right)+F_{34}\! \left(x , y\right)
F_{34}\! \left(x , y\right) = F_{22}\! \left(x , y\right) F_{35}\! \left(x , y\right)
F_{35}\! \left(x , y\right) = F_{36}\! \left(x \right)+F_{39}\! \left(x , y\right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{14}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x , y\right) = F_{24}\! \left(x \right)+F_{34}\! \left(x , y\right)+F_{40}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{41}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = F_{19}\! \left(x , y\right)+F_{39}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{43}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{33}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{45}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{21}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x , 1\right)
F_{48}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{49}\! \left(x , y\right)+F_{50}\! \left(x , y\right)+F_{52}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = F_{22}\! \left(x , y\right) F_{48}\! \left(x , y\right)
F_{50}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{51}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = \frac{F_{48}\! \left(x , y\right) y -F_{48}\! \left(x , 1\right)}{y -1}
F_{52}\! \left(x , y\right) = F_{53}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = \frac{F_{7}\! \left(x , y\right) y -F_{7}\! \left(x , 1\right)}{y -1}
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_14(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_46(x))
Eq(F_4(x), F_5(x, 1))
Eq(F_5(x, k_0), F_6(x, k_0))
Eq(F_6(x, k_0), F_22(x, k_0)*F_7(x, k_0))
Eq(F_7(x, k_0), F_44(x, k_0) + F_8(x, k_0))
Eq(F_8(x, k_0), F_18(x, k_0) + 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_14(x))
Eq(F_12(x), F_13(x) + F_15(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_10(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_14(x))
Eq(F_18(x, k_0), F_19(x, k_0) + F_23(x, k_0))
Eq(F_19(x, k_0), F_20(x, k_0))
Eq(F_20(x, k_0), F_21(x, k_0)*F_22(x, k_0))
Eq(F_21(x, k_0), F_1(x) + F_19(x, k_0))
Eq(F_22(x, k_0), k_0*x)
Eq(F_23(x, k_0), F_24(x) + F_25(x, k_0) + F_27(x, k_0))
Eq(F_24(x), 0)
Eq(F_25(x, k_0), F_22(x, k_0)*F_26(x, k_0))
Eq(F_26(x, k_0), F_10(x) + F_23(x, k_0))
Eq(F_27(x, k_0), F_14(x)*F_28(x, k_0))
Eq(F_28(x, k_0), F_29(x, k_0) + F_32(x, k_0))
Eq(F_29(x, k_0), F_19(x, k_0) + F_30(x, k_0))
Eq(F_30(x, k_0), F_31(x, k_0))
Eq(F_31(x, k_0), F_14(x)*F_19(x, k_0))
Eq(F_32(x, k_0), F_33(x, k_0) + F_42(x, k_0))
Eq(F_33(x, k_0), F_24(x) + F_27(x, k_0) + F_34(x, k_0))
Eq(F_34(x, k_0), F_22(x, k_0)*F_35(x, k_0))
Eq(F_35(x, k_0), F_36(x) + F_39(x, k_0))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_14(x)*F_38(x))
Eq(F_38(x), F_1(x) + F_36(x))
Eq(F_39(x, k_0), F_24(x) + F_34(x, k_0) + F_40(x, k_0))
Eq(F_40(x, k_0), F_14(x)*F_41(x, k_0))
Eq(F_41(x, k_0), F_19(x, k_0) + F_39(x, k_0))
Eq(F_42(x, k_0), F_43(x, k_0))
Eq(F_43(x, k_0), F_14(x)*F_33(x, k_0))
Eq(F_44(x, k_0), F_45(x, k_0))
Eq(F_45(x, k_0), F_14(x)*F_21(x, k_0)*F_9(x))
Eq(F_46(x), F_14(x)*F_47(x))
Eq(F_47(x), F_48(x, 1))
Eq(F_48(x, k_0), F_1(x) + F_49(x, k_0) + F_50(x, k_0) + F_52(x, k_0))
Eq(F_49(x, k_0), F_22(x, k_0)*F_48(x, k_0))
Eq(F_50(x, k_0), F_14(x)*F_51(x, k_0))
Eq(F_51(x, k_0), (-k_0*F_48(x, k_0) + F_48(x, 1))/(1 - k_0))
Eq(F_52(x, k_0), F_53(x, k_0))
Eq(F_53(x, k_0), F_14(x)*F_54(x, k_0))
Eq(F_54(x, k_0), (-k_0*F_7(x, k_0) + F_7(x, 1))/(1 - k_0))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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