0132_0213_1023_1230_1302
Counting sequence:
1, 1, 2, 6, 19, 54, 146, 386, 1012, 2646, 6915, 18076, 47270, 123658, 323574, 846846, 2216603, 5802370, 15189538, 39764666, 104101896, 272536862, 713501947, 1867958056, 4890354534, 12803076914, 33518829866, 87753337686, 229741061827, 601469651406, 1574667574610, 4122532558226, 10792929268060, 28256253899718, 73975830252819, 193671233334196, 507037864046918, 1327442349579130, 3475289169760158, 9098425135543566, 23819986197782411, 62361533394557722, 163264613883556642, 427432308090532106, 1119032310120125424, 2929664621836349774, 7669961554687515211, 20080220041091292736, 52570698566750051142, 137631875656187645666, 360324928397005358930, 943342909527049689126, 2469703800171557439475, 6465768490967257618278, 16927601672697264135314, 44317036527071218496546, 116023507908430123783108, 303753487198079568990390, 795236953685582731754403, 2081957373858303190976716, 5450635167888735554445926, 14269948129806946750335082, 37359209221530556687803462, 97807679534782218582293406, 256063829382812046319538939, 670383808613647362906003634, 1755087596458119432188614306, 4594878980760693765979661786, 12029549345823934087860335832, 31493769056711063552031132926, 82451757824309183844772814875, 215861504416216370313256850776, 565132755424339736702507028390, 1479536761856802531732743064338, 3873477530146067360041710285434, 10140895828581398741876854742646, 26549209955598127560619309013923, 69506734038212981828495994321150, 181970992159040814508414051042898, 476406242438909456168806457922866, 1247247735157687545053610998934316, 3265336963034153164519692514203942, 8548763153944771925088738195209907, 22380952498800162572857459698281956, 58594094342455715732177850178024454, 153401330528566984524481237741035994, 401609897243245237680765219228716526, 1051428361201168728258118923033991086, 2752675186360260946673395409145767147, 7206597197879614111082175666764698186, 18867116407278581385473043812782225570, 49394752023956130043556976355577264426, 129317139664589808742317818059578751680, 338556666969813296178736431212783460398, 886352861244850079786351361774025283179, 2320501916764736943168117493694170512496, 6075152889049360749698260845088618031238, 15904956750383345305894724606936693481410, 41639717362100675167934232266866604090018, 109014195335918680197824351050173270365766, 285402868645655365425403519031308500261331
Generating function in Maple syntax:
-(x^7+x^6-2*x^5-3*x^4-2*x^3+8*x^2-5*x+1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{7}+x^{6}-2 x^{5}-3 x^{4}-2 x^{3}+8 x^{2}-5 x +1}{\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**7 - x**6 + 2*x**5 + 3*x**4 + 2*x**3 - 8*x**2 + 5*x - 1)/((x - 1)**2*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^2+x-1)*(x-1)^2*F(x)+x^7+x^6-2*x^5-3*x^4-2*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{7}+x^{6}-2 x^{5}-3 x^{4}-2 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(-25*5^(1/2)+65)*(3/2-1/2*5^(1/2))^(-n)+1/20*(25*5^(1/2)+65)*(3/2+1/2*5^(1/2))^(-n)+1/20*(5-5^(1/2))*(-1/2*5^(1/2)-1/2)^(-n)+1/20*(5+5^(1/2))*(1/2*5^(1/2)-1/2)^(-n)-n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-25 \sqrt{5}+65\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(25 \sqrt{5}+65\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(5-\sqrt{5}\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\frac{\left(5+\sqrt{5}\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}-n -1 & \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) = 54
a(6) = 146
a(7) = 386
a(n+4) = a(n)-2*a(n+1)-3*a(n+2)+4*a(n+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) = 54
a \! \left(6\right) = 146
a \! \left(7\right) = 386
a \! \left(n +4\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right)-n -1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18299/
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[23,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[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[13,x]
F[23,x] = F[24,x]+F[38,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[24,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[37,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[28,x]
F[36,x] = F[31,x]
F[37,x] = F[12,x]*F[30,x]
F[38,x] = F[32,x]+F[39,x]+F[58,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[45,x]
F[41,x] = F[28,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[12,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[32,x]+F[37,x]+F[39,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[32,x]+F[51,x]+F[57,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[55,x]
F[53,x] = F[12,x]+F[54,x]
F[54,x] = F[43,x]
F[55,x] = F[50,x]+F[56,x]
F[56,x] = F[48,x]
F[57,x] = F[12,x]*F[24,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[63,x]
F[60,x] = F[24,x]+F[61,x]
F[61,x] = F[32,x]+F[51,x]+F[62,x]
F[62,x] = F[12,x]*F[60,x]
F[63,x] = F[64,x]+F[75,x]
F[64,x] = F[32,x]+F[33,x]+F[65,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[72,x]
F[67,x] = F[24,x]+F[68,x]
F[68,x] = F[32,x]+F[57,x]+F[69,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[44,x]+F[71,x]
F[71,x] = F[68,x]
F[72,x] = F[64,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[12,x]*F[64,x]
F[75,x] = F[76,x]
F[76,x] = F[12,x]*F[63,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_{23}\! \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_{21}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{32}\! \left(x \right) = 0
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_{28}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{58}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{28}\! \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_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{32}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{32}\! \left(x \right)+F_{51}\! \left(x \right)+F_{57}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{43}\! \left(x \right)
F_{55}\! \left(x \right) = F_{50}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{48}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = F_{24}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{32}\! \left(x \right)+F_{51}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{75}\! \left(x \right)
F_{64}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \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_{72}\! \left(x \right)
F_{67}\! \left(x \right) = F_{24}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{32}\! \left(x \right)+F_{57}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{44}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{68}\! \left(x \right)
F_{72}\! \left(x \right) = F_{64}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \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_23(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_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_13(x))
Eq(F_23(x), F_24(x) + F_38(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_24(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_37(x))
Eq(F_32(x), 0)
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_28(x))
Eq(F_36(x), F_31(x))
Eq(F_37(x), F_12(x)*F_30(x))
Eq(F_38(x), F_32(x) + F_39(x) + F_58(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_45(x))
Eq(F_41(x), F_28(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_12(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_32(x) + F_37(x) + F_39(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_32(x) + F_51(x) + F_57(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_55(x))
Eq(F_53(x), F_12(x) + F_54(x))
Eq(F_54(x), F_43(x))
Eq(F_55(x), F_50(x) + F_56(x))
Eq(F_56(x), F_48(x))
Eq(F_57(x), F_12(x)*F_24(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_63(x))
Eq(F_60(x), F_24(x) + F_61(x))
Eq(F_61(x), F_32(x) + F_51(x) + F_62(x))
Eq(F_62(x), F_12(x)*F_60(x))
Eq(F_63(x), F_64(x) + F_75(x))
Eq(F_64(x), F_32(x) + F_33(x) + F_65(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_72(x))
Eq(F_67(x), F_24(x) + F_68(x))
Eq(F_68(x), F_32(x) + F_57(x) + F_69(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_44(x) + F_71(x))
Eq(F_71(x), F_68(x))
Eq(F_72(x), F_64(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_12(x)*F_64(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_12(x)*F_63(x))
Pack JSON:
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Specification JSON:
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