0213_1320_2301_3021
Counting sequence:
1, 1, 2, 6, 20, 61, 171, 461, 1222, 3216, 8438, 22111, 57909, 151631, 397000, 1039386, 2721176, 7124161, 18651327, 48829841, 127838218, 334684836, 876216314, 2293964131, 6005676105, 15723064211, 41163516556, 107767485486, 282138939932, 738649334341, 1933809063123, 5062777855061, 13254524502094, 34700795651256, 90847862451710, 237842791703911, 622680512660061, 1630198746276311, 4267915726168912, 11173548432230466, 29252729570522528, 76584640279337161, 200501191267488999, 524918933523129881, 1374255609301900690, 3597847894382572236, 9419288073845816066, 24660016327154876011, 64560760907618812017, 169022266395701560091, 442506038279485868308, 1158495848442756044886, 3032981507048782266404, 7940448672703590754381, 20788364511061989996795, 54424644860482379236061, 142485570070385147711446, 373032065350673063898336, 976610625981634043983622, 2556799812594229068052591, 6693788811801053160174213, 17524566622808930412470111, 45879911056625738077236184, 120115166547068283819238506, 314465588584579113380479400, 823281599206669056322199761, 2155379209035428055586119951, 5642856027899615110436160161, 14773188874663417275722360602, 38676710596090636716730921716, 101256942913608492874470404618, 265094118144734841906680292211, 694025411520596032845570472089, 1816982116417053256630031124131, 4756920937730563737044522900380, 12453780696774637954503537577086, 32604421152593350126466089830956, 85359482761005412424894731915861, 223474027130422887148218105916707, 585062598630263249019759585834341, 1531713768760366859911060651586398, 4010078707650837330713422368924936, 10498522354192145132229206455188494, 27485488354925598065974196996640631, 71957942710584649065693384534733485, 188388339776828349131105956607559911, 493207076619900398327624485287946336, 1291232890082872845851767499256279186, 3380491593628718139227678012480891312, 8850241890803281571831266538186394841, 23170234078781126576266121602078293303, 60660460345540098156967098268048485161, 158811146957839167894635173202067162274, 415772980527977405526938421338153001756, 1088507794626093048686180090812391843090, 2849750403350301740531601851099022527611, 7460743415424812172908625462484675739841, 19532479842924134778194274536355004692011, 51136696113347592161674198146580338336292, 133877608497118641706828319903386010316966, 350496129378008332958810761563577692614708
Generating function in Maple syntax:
-(2*x^6-2*x^4+x^3-5*x^2+4*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{2 x^{6}-2 x^{4}+x^{3}-5 x^{2}+4 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**6 + 2*x**4 - x**3 + 5*x**2 - 4*x + 1)/((x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^2*F(x)+2*x^6-2*x^4+x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+2 x^{6}-2 x^{4}+x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/5*(-39*5^(1/2)+90)*(3/2-1/2*5^(1/2))^(-n)+1/5*(39*5^(1/2)+90)*(3/2+1/2*5^(1/2))^(-n)-3-n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-39 \sqrt{5}+90\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{5}+\frac{\left(39 \sqrt{5}+90\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5}-3-n & \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) = 61
a(6) = 171
a(n+2) = -a(n)+3*a(n+1)+n+4, 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) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 171
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+n +4, \quad n \geq 7
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21399/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[17,x]*F[3,x]
F[3,x] = F[1,x]+F[28,x]+F[4,x]
F[4,x] = F[17,x]*F[5,x]
F[5,x] = F[1,x]+F[13,x]+F[27,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[17,x]*F[19,x]*F[8,x]
F[8,x] = F[22,x]+F[9,x]
F[9,x] = F[10,x]*F[19,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]*F[17,x]
F[12,x] = F[1,x]+F[13,x]+F[18,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[17,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[7,x]
F[17,x] = x
F[18,x] = F[12,x]*F[17,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[17,x]*F[19,x]
F[22,x] = F[23,x]
F[23,x] = F[17,x]*F[24,x]
F[24,x] = F[22,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[17,x]*F[19,x]
F[27,x] = F[17,x]*F[5,x]
F[28,x] = F[17,x]*F[29,x]
F[29,x] = F[1,x]+F[30,x]+F[40,x]+F[44,x]
F[30,x] = F[17,x]*F[31,x]
F[31,x] = F[1,x]+F[18,x]+F[32,x]+F[38,x]+F[6,x]
F[32,x] = F[33,x]
F[33,x] = F[17,x]*F[34,x]
F[34,x] = 2*F[18,x]+F[10,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[17,x]*F[37,x]
F[37,x] = F[16,x]+F[34,x]
F[38,x] = F[39,x]
F[39,x] = F[15,x]*F[17,x]
F[40,x] = F[17,x]*F[41,x]
F[41,x] = F[1,x]+F[30,x]+F[40,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[17,x]*F[19,x]
F[44,x] = F[17,x]*F[45,x]
F[45,x] = F[1,x]+F[18,x]+F[32,x]+F[46,x]+F[48,x]
F[46,x] = F[17,x]*F[47,x]
F[47,x] = F[1,x]+F[18,x]+F[32,x]+F[42,x]+F[46,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[17,x]*F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[1,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[17,x]*F[50,x]
F[54,x] = F[19,x]*F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[17,x]*F[57,x]
F[57,x] = F[1,x]+F[56,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_{17}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{17}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)+F_{27}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{17}\! \left(x \right) F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{22}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{19}\! \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_{17}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = x
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{17}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right) F_{5}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{30}\! \left(x \right) = F_{17}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{38}\! \left(x \right)+F_{6}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{10}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16}\! \left(x \right)+F_{34}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{40}\! \left(x \right) = F_{17}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{17}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{1}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{17}\! \left(x \right) F_{50}\! \left(x \right)
F_{54}\! \left(x \right) = F_{19}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{17}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{1}\! \left(x \right)+F_{56}\! \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_17(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_28(x) + F_4(x))
Eq(F_4(x), F_17(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_13(x) + F_27(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_17(x)*F_19(x)*F_8(x))
Eq(F_8(x), F_22(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_19(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x)*F_17(x))
Eq(F_12(x), F_1(x) + F_13(x) + F_18(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_17(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), x)
Eq(F_18(x), F_12(x)*F_17(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_17(x)*F_19(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_17(x)*F_24(x))
Eq(F_24(x), F_22(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_17(x)*F_19(x))
Eq(F_27(x), F_17(x)*F_5(x))
Eq(F_28(x), F_17(x)*F_29(x))
Eq(F_29(x), F_1(x) + F_30(x) + F_40(x) + F_44(x))
Eq(F_30(x), F_17(x)*F_31(x))
Eq(F_31(x), F_1(x) + F_18(x) + F_32(x) + F_38(x) + F_6(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_17(x)*F_34(x))
Eq(F_34(x), F_10(x) + 2*F_18(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_17(x)*F_37(x))
Eq(F_37(x), F_16(x) + F_34(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_15(x)*F_17(x))
Eq(F_40(x), F_17(x)*F_41(x))
Eq(F_41(x), F_1(x) + F_30(x) + F_40(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_17(x)*F_19(x))
Eq(F_44(x), F_17(x)*F_45(x))
Eq(F_45(x), F_1(x) + F_18(x) + F_32(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_17(x)*F_47(x))
Eq(F_47(x), F_1(x) + F_18(x) + F_32(x) + F_42(x) + F_46(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_17(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_1(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_17(x)*F_50(x))
Eq(F_54(x), F_19(x)*F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_17(x)*F_57(x))
Eq(F_57(x), F_1(x) + F_56(x))
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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "row_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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