0231_1203_1302_3012
Counting sequence:
1, 1, 2, 6, 20, 64, 194, 566, 1612, 4521, 12548, 34564, 94654, 257995, 700455, 1895365, 5113735, 13761550, 36949186, 99004735, 264795596, 707045418, 1885097277, 5019147641, 13347157386, 35453335003, 94075534340, 249393397443, 660564773312, 1748228796529, 4623392923781, 12218776180301, 32271667523296, 85184836908987, 224734104366449, 592595663962582, 1561875164353999, 4114781781056968, 10836101165799744, 28525752734983399, 75067182034672861, 197479631218867027, 519353631099566290, 1365467825727791428, 3589098016836950771, 9431531847988747497, 24778738857130202217, 65085266439636495551, 170922297546330336471, 448779389946388270561, 1178122487006918150033, 3092261422449185250396, 8115115403510596779497, 21293652368030597470507, 55865863958647826481359, 146550551509949320992650, 384393830028856715472196, 1008130103171264838777636, 2643692776581340276453340, 6932061710358115563109789, 18174962297936364365082115, 47648226429586818905100226, 124906312709506746798623421, 327407498173314906276187729, 858146552852958163048475542, 2249079499339780886361506344, 5894139805193813209268685643, 15445785774707827649969387398, 40473903669519865719052413425, 106051584123963995017913340750, 277867391085268548211099384305, 728010522664636825502643283196, 1907298175795677163995981933414, 4996679959755249016474851358576, 13089635329764696634552785578525, 34289222756214181840830069316170, 89819939579204642006871257467718, 235273919788214092134658864088137, 616256571991087083482876495127524, 1614123905894553594239480962902491, 4227663794922795241373377441358701, 11072685783864134466061657325237982, 28999807859608229098623094176540206, 75949949429454189542562027096133882, 198907270367006688679566114360131647, 520912966160922952359811314445137379, 1364179531285409636240315392252685232, 3572483407741867495856059504862962984, 9355385609746272897696348871531984234, 24498887901997406088753649583598099178, 64154134769357049321289594956393979000, 167995215449434123367517260472372291208, 439909667828332691661807699243813704306, 1151926487835644839536252247688884786779, 3016344910762069668694813497187015824138, 7898279674532193206589550543529055957758, 20681382357143929566255004296496297188204, 54152988568979532277936908860152475302645, 141795141106812203454819764424221641619645, 371275724652055797752314818891879738089823, 972138767182902197886258559010162081367293
Generating function in Maple syntax:
-(2*x-1)*(4*x^6-14*x^5+27*x^4-30*x^3+20*x^2-7*x+1)/(x^2-3*x+1)/(3*x^3-5*x^2+4*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(4 x^{6}-14 x^{5}+27 x^{4}-30 x^{3}+20 x^{2}-7 x +1\right)}{\left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(1 - 2*x)*(4*x**6 - 14*x**5 + 27*x**4 - 30*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**3*(x**2 - 3*x + 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(3*x^3-5*x^2+4*x-1)*(x-1)^3*F(x)+(2*x-1)*(4*x^6-14*x^5+27*x^4-30*x^3+20*x^2-7*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(2 x -1\right) \left(4 x^{6}-14 x^{5}+27 x^{4}-30 x^{3}+20 x^{2}-7 x +1\right) = 0
Explicit closed form in Maple syntax:
1/450120*(3410*((-48/31*I+16/31*3^(1/2))*31^(1/2)+I*3^(1/2)-1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)-150040+5425*2^(1/3)*((-129/1085*I-43/1085*3^(1/2))*31^(1/2)+I*3^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(47/8712*2^(1/3)*((I-9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/450120*(-3410*((-48/31*I-16/31*3^(1/2))*31^(1/2)+I*3^(1/2)+1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)-150040-5425*2^(1/3)*((-129/1085*I+43/1085*3^(1/2))*31^(1/2)+I*3^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(-47/8712*((I+9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)-1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)+1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/450120*((-3520*31^(1/2)*2^(2/3)*3^(1/2)+6820*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)-150040+(430*31^(1/2)*3^(1/2)*2^(1/3)-10850*2^(1/3))*(47+9*31^(1/2)*3^(1/2))^(2/3))*(1/4356*2^(1/3)*(9*31^(1/2)*3^(1/2)-47)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/18*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/450120*(-90024*5^(1/2)+900240)*(3/2-1/2*5^(1/2))^(-n)+1/450120*(90024*5^(1/2)+900240)*(3/2+1/2*5^(1/2))^(-n)-1/2*n^2+3/2*n-2
Explicit closed form in latex syntax:
\frac{\left(3410 \left(\left(-\frac{48 \,\mathrm{I}}{31}+\frac{16 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-150040+5425 \,2^{\frac{1}{3}} \left(\left(-\frac{129 \,\mathrm{I}}{1085}-\frac{43 \sqrt{3}}{1085}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(-3410 \left(\left(-\frac{48 \,\mathrm{I}}{31}-\frac{16 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-150040-5425 \,2^{\frac{1}{3}} \left(\left(-\frac{129 \,\mathrm{I}}{1085}+\frac{43 \sqrt{3}}{1085}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(\left(-3520 \sqrt{31}\, 2^{\frac{2}{3}} \sqrt{3}+6820 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-150040+\left(430 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}-10850 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(-90024 \sqrt{5}+900240\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{450120}+\frac{\left(90024 \sqrt{5}+900240\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{450120}-\frac{n^{2}}{2}+\frac{3 n}{2}-2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 194
a(7) = 566
a(n+5) = -1/2*n^2+3*a(n)-14*a(n+1)+22*a(n+2)-18*a(n+3)+7*a(n+4)+1/2*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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 194
a \! \left(7\right) = 566
a \! \left(n +5\right) = -\frac{n^{2}}{2}+3 a \! \left(n \right)-14 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right)+\frac{n}{2}-1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15890/
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[13,x]*F[4,x]
F[4,x] = F[31,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[14,x]+F[6,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = F[20,x]+F[24,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[23,x]
F[23,x] = F[21,x]+F[6,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[10,x]+F[25,x]
F[28,x] = F[29,x]
F[29,x] = F[13,x]*F[30,x]
F[30,x] = F[21,x]+F[28,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[15,x]+F[33,x]+F[37,x]
F[33,x] = F[13,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[6,x]
F[36,x] = F[32,x]
F[37,x] = F[13,x]*F[38,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[2,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[13,x]*F[42,x]
F[42,x] = F[35,x]+F[43,x]
F[43,x] = F[40,x]
F[44,x] = F[45,x]+F[52,x]
F[45,x] = F[15,x]+F[46,x]+F[50,x]
F[46,x] = F[13,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[10,x]
F[49,x] = F[45,x]
F[50,x] = F[13,x]*F[51,x]
F[51,x] = F[2,x]+F[45,x]
F[52,x] = F[53,x]
F[53,x] = F[13,x]*F[54,x]
F[54,x] = F[40,x]+F[52,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{31}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \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_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \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_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{2}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{45}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{13}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{40}\! \left(x \right)+F_{52}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_31(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_14(x) + F_6(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_24(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_23(x))
Eq(F_23(x), F_21(x) + F_6(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_10(x) + F_25(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_13(x)*F_30(x))
Eq(F_30(x), F_21(x) + F_28(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_15(x) + F_33(x) + F_37(x))
Eq(F_33(x), F_13(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_6(x))
Eq(F_36(x), F_32(x))
Eq(F_37(x), F_13(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_2(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_13(x)*F_42(x))
Eq(F_42(x), F_35(x) + F_43(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_45(x) + F_52(x))
Eq(F_45(x), F_15(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_13(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_10(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_13(x)*F_51(x))
Eq(F_51(x), F_2(x) + F_45(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_13(x)*F_54(x))
Eq(F_54(x), F_40(x) + F_52(x))
Pack JSON:
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Specification JSON:
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