0213_0231_2130_2301
Counting sequence:
1, 1, 2, 6, 20, 63, 183, 497, 1281, 3170, 7598, 17758, 40682, 91721, 204145, 449623, 981743, 2128172, 4585216, 9827420, 20967728, 44559723, 94365947, 199222221, 419421645, 880793318, 1845481218, 3858744842, 8053046246, 16777195645, 34896585653, 72477545843, 150323824027, 311385093136, 644245053620, 1331439815528, 2748779017228, 5669356771967, 11682310979231, 24051816783945, 49478023167833, 101704825478058, 208907209176342, 428809534720886, 879609302097570, 1803199069417073, 3694359069178553, 7564639998935887, 15481123718907911, 31665934879754420, 64739244643239208, 132293239053778228, 270215977641980072, 551690954352615251, 1125899906842331395, 2296835809958636933, 4683743612464975013, 9547631210025084462, 19455550390240147946, 39631676720859940770, 80704505322478833438, 164291314406475206693, 334347236335985101437, 680223687718039158251, 1383505805528215775795, 2813128471240705986328, 5718490662849960324188, 11621448766437016797632, 23611832414348225302276, 47961534591644833387495, 97398808709186431668263, 197749096470166392406913, 401401151043919842192817, 814608218295013798334066, 1652828269004375823705470, 3352880202837448100573838, 6800207735332289106507098, 13789310129979364022709657, 27956409578588299663727361, 56668397794435742562925895, 114847952863389771595584543, 232718220275816116129357628, 471481069649705378133745232, 955051397495557048016130316, 1934281311383406679528044320, 3916919655551398526046081083, 7930553376671967386070490123, 16054534884482275440095894077, 32495926031241232216099785341, 65765564587035827104013642870, 133078554223178379551653412818, 269251958544570209790556963898, 544693617285567320955611986262, 1101766634963988444660217765581, 2228292070713684494818420683845, 4506101742998784200632809126243, 9111238689140398823257551105483, 18420547784566458490498965131552, 37237236381704238668965653193476, 75266754388551120713866749207320, 152118072027387528179604380881148
Generating function in Maple syntax:
(2*x^8-9*x^6+29*x^5-49*x^4+48*x^3-27*x^2+8*x-1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
\frac{2 x^{8}-9 x^{6}+29 x^{5}-49 x^{4}+48 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(2*x**8 - 9*x**6 + 29*x**5 - 49*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^5*F(x)+2*x^8-9*x^6+29*x^5-49*x^4+48*x^3-27*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+2 x^{8}-9 x^{6}+29 x^{5}-49 x^{4}+48 x^{3}-27 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,3+5/4*2^n*n-5*2^n-1/24*n^4+5/12*n^3-35/24*n^2+73/12*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 3+\frac{5 \,2^{n} n}{4}-5 \,2^{n}-\frac{n^{4}}{24}+\frac{5 n^{3}}{12}-\frac{35 n^{2}}{24}+\frac{73 n}{12} & \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) = 63
a(6) = 183
a(7) = 497
a(8) = 1281
a(n+2) = -4*a(n)+4*a(n+1)-1/24*(-1+n)*(n^3-17*n^2+78*n-192), n >= 9
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) = 63
a \! \left(6\right) = 183
a \! \left(7\right) = 497
a \! \left(8\right) = 1281
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{\left(-1+n \right) \left(n^{3}-17 n^{2}+78 n -192\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21356/
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[15,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[15,x]*F[7,x]
F[7,x] = F[59,x]+F[8,x]
F[8,x] = F[54,x]*F[9,x]
F[9,x] = F[10,x]+F[47,x]
F[10,x] = F[11,x]+F[19,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]
F[15,x] = x
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[15,x]*F[16,x]
F[19,x] = F[20,x]
F[20,x] = F[15,x]*F[21,x]
F[21,x] = F[22,x]+F[30,x]
F[22,x] = F[23,x]
F[23,x] = F[15,x]*F[24,x]
F[24,x] = F[12,x]+F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[15,x]*F[28,x]
F[28,x] = F[17,x]+F[29,x]
F[29,x] = F[27,x]
F[30,x] = F[31,x]+F[32,x]+F[44,x]
F[31,x] = 0
F[32,x] = F[15,x]*F[33,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[13,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[15,x]*F[34,x]
F[37,x] = F[30,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[15,x]*F[40,x]
F[40,x] = F[41,x]+F[46,x]
F[41,x] = F[31,x]+F[42,x]+F[44,x]
F[42,x] = F[15,x]*F[43,x]
F[43,x] = F[13,x]+F[41,x]
F[44,x] = F[15,x]*F[45,x]
F[45,x] = F[17,x]+F[41,x]
F[46,x] = F[39,x]
F[47,x] = F[48,x]
F[48,x] = F[15,x]*F[16,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[15,x]*F[52,x]
F[52,x] = F[12,x]+F[21,x]
F[53,x] = F[13,x]*F[17,x]
F[54,x] = F[1,x]+F[55,x]
F[55,x] = F[15,x]*F[56,x]
F[56,x] = F[54,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[55,x]
F[59,x] = F[60,x]+F[69,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[17,x]+F[62,x]
F[62,x] = F[31,x]+F[63,x]+F[66,x]
F[63,x] = F[15,x]*F[64,x]
F[64,x] = F[58,x]+F[65,x]
F[65,x] = F[63,x]
F[66,x] = F[15,x]*F[67,x]
F[67,x] = F[61,x]+F[68,x]
F[68,x] = F[62,x]
F[69,x] = F[50,x]*F[70,x]
F[70,x] = F[17,x]+F[71,x]
F[71,x] = F[31,x]+F[72,x]+F[74,x]
F[72,x] = F[15,x]*F[73,x]
F[73,x] = F[58,x]
F[74,x] = F[15,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[17,x]
F[77,x] = F[71,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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{59}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{54}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{47}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{44}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{15}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right) F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{41}\! \left(x \right) = F_{31}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{17}\! \left(x \right)+F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = F_{39}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right)+F_{21}\! \left(x \right)
F_{53}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{54}\! \left(x \right) = F_{1}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{15}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{55}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{69}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{17}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{31}\! \left(x \right)+F_{63}\! \left(x \right)+F_{66}\! \left(x \right)
F_{63}\! \left(x \right) = F_{15}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{58}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{15}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{61}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{62}\! \left(x \right)
F_{69}\! \left(x \right) = F_{50}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{17}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{31}\! \left(x \right)+F_{72}\! \left(x \right)+F_{74}\! \left(x \right)
F_{72}\! \left(x \right) = F_{15}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{58}\! \left(x \right)
F_{74}\! \left(x \right) = F_{15}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{17}\! \left(x \right)
F_{77}\! \left(x \right) = F_{71}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_15(x)*F_7(x))
Eq(F_7(x), F_59(x) + F_8(x))
Eq(F_8(x), F_54(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_47(x))
Eq(F_10(x), F_11(x) + F_19(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x))
Eq(F_15(x), x)
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_15(x)*F_16(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_15(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_30(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_15(x)*F_24(x))
Eq(F_24(x), F_12(x) + F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_15(x)*F_28(x))
Eq(F_28(x), F_17(x) + F_29(x))
Eq(F_29(x), F_27(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_44(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_15(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_13(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_15(x)*F_34(x))
Eq(F_37(x), F_30(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_15(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_46(x))
Eq(F_41(x), F_31(x) + F_42(x) + F_44(x))
Eq(F_42(x), F_15(x)*F_43(x))
Eq(F_43(x), F_13(x) + F_41(x))
Eq(F_44(x), F_15(x)*F_45(x))
Eq(F_45(x), F_17(x) + F_41(x))
Eq(F_46(x), F_39(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_15(x)*F_16(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_15(x)*F_52(x))
Eq(F_52(x), F_12(x) + F_21(x))
Eq(F_53(x), F_13(x)*F_17(x))
Eq(F_54(x), F_1(x) + F_55(x))
Eq(F_55(x), F_15(x)*F_56(x))
Eq(F_56(x), F_54(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_55(x))
Eq(F_59(x), F_60(x) + F_69(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_17(x) + F_62(x))
Eq(F_62(x), F_31(x) + F_63(x) + F_66(x))
Eq(F_63(x), F_15(x)*F_64(x))
Eq(F_64(x), F_58(x) + F_65(x))
Eq(F_65(x), F_63(x))
Eq(F_66(x), F_15(x)*F_67(x))
Eq(F_67(x), F_61(x) + F_68(x))
Eq(F_68(x), F_62(x))
Eq(F_69(x), F_50(x)*F_70(x))
Eq(F_70(x), F_17(x) + F_71(x))
Eq(F_71(x), F_31(x) + F_72(x) + F_74(x))
Eq(F_72(x), F_15(x)*F_73(x))
Eq(F_73(x), F_58(x))
Eq(F_74(x), F_15(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_17(x))
Eq(F_77(x), F_71(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": 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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