0132_1230_2103_2130_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 114, 231, 448, 855, 1627, 3108, 5976, 11572, 22554, 44200, 87005, 171861, 340394, 675601, 1343064, 2673281, 5326171, 10619830, 21187632, 42291766, 84449234, 168682106, 337015213, 673466975, 1346023682, 2690576123, 5378773536, 10753700267, 21501178539, 42992292200, 85968301880, 171910261176, 343777902554, 687486848556, 1374862127149, 2749543734761, 5498795387626, 10997118182085, 21993471698072, 43985706146581, 87969410387979, 175935581632506, 351865922228528, 703723364290170, 1407433007390962, 2814843813440638, 5629651704366621, 11259245284894387, 22518396523453474, 45036640876752175, 90073035537034880, 180145672687467391, 360290700771886139, 720580358554145836, 1441159029515642584, 2882315328449037116, 5764626238723206106, 11529245328689326640, 23058479090446731837, 46116939465204490621, 92233848647788122474, 184467648297266450681, 368935217313602287768, 737870306347964209065, 1475740405135757481467, 2951480474432103705278, 5902960405464625263856, 11805919931690257173182, 23611838440741938897234, 47223674579606309045250, 94447345434696473949005, 188894684842999235066631, 377789359935088613222274, 755578704092873656766691, 1511157382657533887011744, 3022314724009550777893971, 6044629381185371133209611, 12089258654231494847787120, 24178517133490011854442552, 48357033983867798449203136, 96714067509650393797677594, 193428134278103359235032756, 386856267356924087009105933, 773712532773368114197024785, 1547425062406973537112001322, 3094850119733704323120869293, 6189700231247403203856662616, 12379400449194558214225225277, 24758800876868862792577387819, 49517601718917223755793730050, 99035203381493692046353472880, 198070406671826126798112038146, 396140813196150240836395310834, 792281626153637211618367082246, 1584563251921109140422481997629
Generating function in Maple syntax:
(4*x^11-2*x^10-10*x^9+17*x^7-6*x^6+2*x^4-13*x^3+14*x^2-6*x+1)/(2*x-1)/(x^2+x-1)/(x-1)^4
Generating function in latex syntax:
\frac{4 x^{11}-2 x^{10}-10 x^{9}+17 x^{7}-6 x^{6}+2 x^{4}-13 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(4*x**11 - 2*x**10 - 10*x**9 + 17*x**7 - 6*x**6 + 2*x**4 - 13*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(2*x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2+x-1)*(x-1)^4*F(x)+4*x^11-2*x^10-10*x^9+17*x^7-6*x^6+2*x^4-13*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{11}-2 x^{10}-10 x^{9}+17 x^{7}-6 x^{6}+2 x^{4}-13 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
1/6*n^3-1/2*n^2+piecewise(n = 0,63/4,n = 1,25/2,n = 2,11,n = 3,6,n = 4,2,0)-2/3*n-17-7/10*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+7/10*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+1/2*(-1/2*5^(1/2)-1/2)^(-n)+1/2*(1/2*5^(1/2)-1/2)^(-n)+5/4*2^n
Explicit closed form in latex syntax:
\frac{n^{3}}{6}-\frac{n^{2}}{2}+\left(\left\{\begin{array}{cc}\frac{63}{4} & n =0 \\ \frac{25}{2} & n =1 \\ 11 & n =2 \\ 6 & n =3 \\ 2 & n =4 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)-\frac{2 n}{3}-17-\frac{7 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{10}+\frac{7 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{10}+\frac{\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2}+\frac{\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2}+\frac{5 \,2^{n}}{4}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 114
a(7) = 231
a(8) = 448
a(9) = 855
a(10) = 1627
a(11) = 3108
a(n+3) = 1/6*n^3-3/2*n^2-2*a(n)-a(n+1)+3*a(n+2)+1/3*n-14, n >= 12
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) = 51
a \! \left(6\right) = 114
a \! \left(7\right) = 231
a \! \left(8\right) = 448
a \! \left(9\right) = 855
a \! \left(10\right) = 1627
a \! \left(11\right) = 3108
a \! \left(n +3\right) = \frac{n^{3}}{6}-\frac{3 n^{2}}{2}-2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{n}{3}-14, \quad n \geq 12
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/20117/
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[31,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[21,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[20,x]*F[4,x]
F[20,x] = F[14,x]
F[21,x] = F[22,x]+F[23,x]+F[28,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[14,x]+F[30,x]
F[30,x] = F[23,x]
F[31,x] = F[32,x]+F[42,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[1,x]+F[14,x]
F[36,x] = F[32,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[32,x]
F[41,x] = F[37,x]
F[42,x] = F[22,x]+F[43,x]+F[60,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[46,x]+F[7,x]
F[46,x] = F[22,x]+F[23,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[29,x]+F[49,x]
F[49,x] = F[50,x]+F[54,x]
F[50,x] = F[22,x]+F[23,x]+F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[20,x]+F[53,x]
F[53,x] = F[50,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[26,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[57,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[67,x]
F[62,x] = F[32,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[11,x]+F[30,x]
F[67,x] = F[68,x]+F[75,x]
F[68,x] = F[22,x]+F[69,x]+F[72,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[14,x]+F[50,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[40,x]+F[74,x]
F[74,x] = F[68,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]
F[78,x] = F[21,x]+F[54,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_{31}\! \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_{21}\! \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_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{42}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{22}\! \left(x \right)+F_{43}\! \left(x \right)+F_{60}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{7}\! \left(x \right)
F_{46}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{50}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{26}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{67}\! \left(x \right)
F_{62}\! \left(x \right) = F_{32}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{75}\! \left(x \right)
F_{68}\! \left(x \right) = F_{22}\! \left(x \right)+F_{69}\! \left(x \right)+F_{72}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{14}\! \left(x \right)+F_{50}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{40}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{68}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{21}\! \left(x \right)+F_{54}\! \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_31(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_21(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_20(x)*F_4(x))
Eq(F_20(x), F_14(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_28(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_14(x) + F_30(x))
Eq(F_30(x), F_23(x))
Eq(F_31(x), F_32(x) + F_42(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_1(x) + F_14(x))
Eq(F_36(x), F_32(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_32(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), F_22(x) + F_43(x) + F_60(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_46(x) + F_7(x))
Eq(F_46(x), F_22(x) + F_23(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_29(x) + F_49(x))
Eq(F_49(x), F_50(x) + F_54(x))
Eq(F_50(x), F_22(x) + F_23(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_20(x) + F_53(x))
Eq(F_53(x), F_50(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_26(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_57(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_67(x))
Eq(F_62(x), F_32(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_11(x) + F_30(x))
Eq(F_67(x), F_68(x) + F_75(x))
Eq(F_68(x), F_22(x) + F_69(x) + F_72(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_14(x) + F_50(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_40(x) + F_74(x))
Eq(F_74(x), F_68(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_21(x) + F_54(x))
Pack JSON:
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Specification JSON:
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