0132_1032_1203_2130_3102
Counting sequence:
1, 1, 2, 6, 19, 52, 132, 326, 803, 1993, 4996, 12637, 32194, 82467, 212129, 547438, 1416446, 3672698, 9539560, 24814128, 64623363, 168465035, 439525268, 1147487831, 2997431474, 7833270277, 20478259857, 53551175096, 140070404958, 366441932748, 958802404856, 2509025753722, 6566328694819, 17185933793997, 44983151182644, 117746339878833, 308220434967186, 806841950571623, 2112155092884625, 5529314090428578, 14475151523232238, 37894834809200350, 99206672844765528, 259719686166945956, 679941115661875379, 1780080571074483727, 4660253318561567332, 12200582627586958891, 31941296652106293538, 83622902708452882953, 218926584640508249441, 573155162363211581932, 1500535454380661770094, 3928444163904355939952, 10284782682108593782984, 26925874609023847476078, 70492781472267885960051, 184552348210589871373073, 483164015461522222752740, 1264939193770818531358853, 3311652539030218330818338, 8670016333649609361151915, 22698392210520561514853313, 59425151651000790632018006, 155577045160428865126744734, 407305948089719161935918402, 1066340726474673830449607432, 2791716083760349739736445336, 7308807225046784190979467043, 19134704982637448400786702483, 50095306486933708072840629876, 131151211969406481793236112767, 343358324329984372963031405394, 898923750690369955818509138957, 2353412906785624226758445313649, 6161314927165204378630019814048, 16130531788526800596760588911166, 42230280263687637545474873942548, 110560308648358625824440641816280, 289450644963588534531660978948578, 757791624787918101046691569132163, 1983924226453409083297755115019605, 5193981048603237914499342225729748, 13598018907267046464053979815019481, 35600075648717153630466356431436914, 93202207989318455823145293316242927, 244006548218897370810955302016580657, 638817436464274118914462911382759306, 1672445760762890207263459512731491598, 4378519844992655538981050754613756838, 11463113772532251668776128939918668536, 30010821469199764359312540309169539852, 78569350628180999940636564208463451219, 205697230401416410020636624223605540887, 538522340547905094227530907208397965764, 1409869791185353630854828604756228508243, 3691087032893027374683414541494668867906, 9663391307260995765809763528046742598673, 25299086888419542707817431538183891575745, 66233869357046894407471359035382404572628, 173402521180799857573625735381319881189838
Generating function in Maple syntax:
(x^10-9*x^9+30*x^8-51*x^7+65*x^6-99*x^5+124*x^4-95*x^3+42*x^2-10*x+1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{x^{10}-9 x^{9}+30 x^{8}-51 x^{7}+65 x^{6}-99 x^{5}+124 x^{4}-95 x^{3}+42 x^{2}-10 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**10 - 9*x**9 + 30*x**8 - 51*x**7 + 65*x**6 - 99*x**5 + 124*x**4 - 95*x**3 + 42*x**2 - 10*x + 1)/((x - 1)**4*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(2*x-1)^2*(x-1)^4*F(x)+x^10-9*x^9+30*x^8-51*x^7+65*x^6-99*x^5+124*x^4-95*x^3+42*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+x^{10}-9 x^{9}+30 x^{8}-51 x^{7}+65 x^{6}-99 x^{5}+124 x^{4}-95 x^{3}+42 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/240*(-24*5^(1/2)+120)*(3/2-1/2*5^(1/2))^(-n)+1/240*(24*5^(1/2)+120)*(3/2+1/2*5^(1/2))^(-n)+1/240*(15*n+15)*2^n+1/6*n^3-1/2*n^2-2/3*n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-24 \sqrt{5}+120\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{240}+\frac{\left(24 \sqrt{5}+120\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{240}+\\\frac{\left(15 n +15\right) 2^{n}}{240}+\frac{n^{3}}{6}-\frac{n^{2}}{2}-\frac{2 n}{3}+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) = 52
a(6) = 132
a(7) = 326
a(8) = 803
a(9) = 1993
a(10) = 4996
a(n+4) = -1/6*n^3+n^2-4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)+13/6*n-5, n >= 11
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) = 52
a \! \left(6\right) = 132
a \! \left(7\right) = 326
a \! \left(8\right) = 803
a \! \left(9\right) = 1993
a \! \left(10\right) = 4996
a \! \left(n +4\right) = -\frac{n^{3}}{6}+n^{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+\frac{13 n}{6}-5, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19704/
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[29,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[20,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[19,x]+F[21,x]+F[22,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[22,x]
F[29,x] = F[30,x]+F[44,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[1,x]+F[14,x]
F[34,x] = F[30,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[30,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[35,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[41,x]
F[44,x] = F[21,x]+F[45,x]+F[53,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[11,x]+F[20,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[11,x]+F[28,x]
F[52,x] = F[49,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[38,x]+F[55,x]
F[55,x] = F[56,x]+F[68,x]
F[56,x] = F[21,x]+F[57,x]+F[63,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[11,x]+F[60,x]
F[60,x] = F[19,x]+F[21,x]+F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[24,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[38,x]+F[65,x]
F[65,x] = F[56,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[65,x]
F[68,x] = F[67,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_{29}\! \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_{20}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{19}\! \left(x \right)+F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{22}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{44}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{21}\! \left(x \right)+F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
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_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right)+F_{20}\! \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_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{52}\! \left(x \right) = F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{38}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{68}\! \left(x \right)
F_{56}\! \left(x \right) = F_{21}\! \left(x \right)+F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{19}\! \left(x \right)+F_{21}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{24}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{38}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{56}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{67}\! \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_29(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_20(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_19(x) + F_21(x) + F_22(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_22(x))
Eq(F_29(x), F_30(x) + F_44(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_1(x) + F_14(x))
Eq(F_34(x), F_30(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_30(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_35(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_41(x))
Eq(F_44(x), F_21(x) + F_45(x) + F_53(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_11(x) + F_20(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_11(x) + F_28(x))
Eq(F_52(x), F_49(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_38(x) + F_55(x))
Eq(F_55(x), F_56(x) + F_68(x))
Eq(F_56(x), F_21(x) + F_57(x) + F_63(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_11(x) + F_60(x))
Eq(F_60(x), F_19(x) + F_21(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_24(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_38(x) + F_65(x))
Eq(F_65(x), F_56(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_65(x))
Eq(F_68(x), F_67(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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