0132_0321_1230_1302_3021
Counting sequence:
1, 1, 2, 6, 19, 52, 132, 328, 819, 2072, 5308, 13722, 35677, 93066, 243204, 636142, 1664717, 4357388, 11406694, 29861792, 78177613, 204669792, 535830302, 1402819426, 3672626039, 9615056482, 25172540902, 65902563398, 172535146119, 451702871412, 1182573464168, 3096017516712, 8105479081127, 21220419721336, 55555780077024, 145446920503322, 380784981425937, 996908023766858, 2609939089866344, 6832909245823182, 17888788647593473, 46833456696946732, 122611581443235402, 321001287632747296, 840392281454993409, 2200175556732218912, 5760134388741648322, 15080227609492710018, 39480548439736464619, 103361417709716665602, 270603704689413512778, 708449696358523852102, 1854745384386158021627, 4855786456799950189556, 12712613986013692522444, 33282055501241127351752, 87133552517709689505307, 228118602051887941135128, 597222253637954133869444, 1563548158861974460440922, 4093422222947969247419333, 10716718509981933281781322, 28056733306997830597887052, 73453481411011558511840366, 192303710926036844937592629, 503457651367098976300894092, 1318069243175260083965044142, 3450750078158681275594190688, 9034180991300783742817478069, 23651792895743669952858191392, 61921197695930226115757041638, 162111800192047008394412876642, 424414202880210799067481528927, 1111130808448585388808031648226, 2908978222465545367356613351214, 7615803858948050713261808338182, 19938433354378606772428811593327, 52199496204187769604024626368948, 136660055258184702039645067437744, 357780669570366336514910575865512, 936681953452914307505086660076943, 2452265190788376586000349404280312, 6420113618912215450495961552675752, 16808075665948269765487535253655386, 44004113378932593845966644208195449, 115204264470849511772412397370832522, 301608680033615941471270547904200112, 789621775629998312641399246341662158, 2067256646856378996452927191120676969, 5412148164939138676717382327020255532, 14169187847961037033699219789939972498, 37095415378943972424380277042799540832, 97117058288870880239441611338458524777, 254255759487668668293944556972575904096, 665650220174135124642392059579269053834, 1742694901034736705633231621765231119362, 4562434482930074992257302805716424161747, 11944608547755488271138676795384041218818, 31271391160336389821158727580435699342994, 81869564933253681192337505945923056653702, 214337303639424653755853790257333470456803
Generating function in Maple syntax:
-(x^7-5*x^6+7*x^5-8*x^4+15*x^3-14*x^2+6*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
-\frac{x^{7}-5 x^{6}+7 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-x**7 + 5*x**6 - 7*x**5 + 8*x**4 - 15*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^4*F(x)+x^7-5*x^6+7*x^5-8*x^4+15*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{7}-5 x^{6}+7 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/30*(18*5^(1/2)-30)*(3/2-1/2*5^(1/2))^(-n)+1/30*(-18*5^(1/2)-30)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-19/6*n+5)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(18 \sqrt{5}-30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-18 \sqrt{5}-30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-\frac{19 n}{6}+5 & \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) = 328
a(n+2) = -1/6*n^3-1/2*n^2-a(n)+3*a(n+1)+11/3*n-1, n >= 8
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) = 328
a \! \left(n +2\right) = -\frac{n^{3}}{6}-\frac{n^{2}}{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{11 n}{3}-1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19478/
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[24,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[12,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[12,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[25,x]+F[44,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[16,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[18,x]+F[33,x]+F[37,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[29,x]
F[36,x] = F[32,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[25,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[12,x]
F[43,x] = F[39,x]
F[44,x] = F[18,x]+F[45,x]+F[76,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[48,x]+F[6,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[20,x]
F[50,x] = F[44,x]+F[51,x]
F[51,x] = 2*F[18,x]+F[52,x]+F[56,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[48,x]
F[55,x] = F[51,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[71,x]
F[58,x] = F[18,x]+F[59,x]+F[69,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[61,x]+F[63,x]
F[61,x] = F[10,x]+F[62,x]
F[62,x] = F[49,x]
F[63,x] = F[58,x]+F[64,x]
F[64,x] = 2*F[18,x]+F[56,x]+F[65,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[62,x]
F[68,x] = F[64,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[25,x]+F[58,x]
F[71,x] = F[72,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[21,x]
F[75,x] = F[71,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[70,x]+F[78,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = F[40,x]
F[80,x] = 2*F[18,x]+F[72,x]+F[81,x]
F[81,x] = F[12,x]*F[78,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_{24}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{44}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
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_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{76}\! \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_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{6}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{50}\! \left(x \right) = F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{48}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{71}\! \left(x \right)
F_{58}\! \left(x \right) = F_{18}\! \left(x \right)+F_{59}\! \left(x \right)+F_{69}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{10}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{49}\! \left(x \right)
F_{63}\! \left(x \right) = F_{58}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{62}\! \left(x \right)
F_{68}\! \left(x \right) = F_{64}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{25}\! \left(x \right)+F_{58}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{21}\! \left(x \right)
F_{75}\! \left(x \right) = F_{71}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{70}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{79}\! \left(x \right) = F_{40}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{72}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \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_24(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_12(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_12(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_25(x) + F_44(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_16(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_18(x) + F_33(x) + F_37(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_29(x))
Eq(F_36(x), F_32(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_25(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_12(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_18(x) + F_45(x) + F_76(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_48(x) + F_6(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_20(x))
Eq(F_50(x), F_44(x) + F_51(x))
Eq(F_51(x), 2*F_18(x) + F_52(x) + F_56(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_48(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_71(x))
Eq(F_58(x), F_18(x) + F_59(x) + F_69(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_63(x))
Eq(F_61(x), F_10(x) + F_62(x))
Eq(F_62(x), F_49(x))
Eq(F_63(x), F_58(x) + F_64(x))
Eq(F_64(x), 2*F_18(x) + F_56(x) + F_65(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_62(x))
Eq(F_68(x), F_64(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_25(x) + F_58(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_21(x))
Eq(F_75(x), F_71(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_70(x) + F_78(x))
Eq(F_78(x), F_79(x) + F_80(x))
Eq(F_79(x), F_40(x))
Eq(F_80(x), 2*F_18(x) + F_72(x) + F_81(x))
Eq(F_81(x), F_12(x)*F_78(x))
Pack JSON:
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Specification JSON:
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