0213_0321_1320_2130
Counting sequence:
1, 1, 2, 6, 20, 66, 209, 634, 1855, 5277, 14697, 40293, 109186, 293329, 782996, 2080128, 5506418, 14537292, 38301911, 100761484, 264768781, 695117511, 1823729247, 4782361431, 12535937680, 32850617131, 86066245034, 225448780914, 590481423920, 1546398143622, 4049518312877, 10603767407278, 27765005133931, 72697690444929, 190340951102181, 498350932664793, 1304763386499118, 3416042306046997, 8943569690071376, 23415079081026804, 61302492186729086, 160494046746601296, 420182946587957267, 1100061390087036256, 2880014417812683865, 7540008251630080971, 19740063113635691259, 51680286642393258219, 135301007919776615260, 354223159329401652367, 927369314493358472582, 2427886473000534028038, 6356293482207964138076, 16641000729022799440554, 43566722215659316293641, 114059192939552913661858, 298610910646194953136343, 781773647085423002637477, 2046710246782856168558241, 5358357525608709730603149, 14028363194734401478384666, 36726733787976751614819481, 96151841627960367186612812, 251728798013433377586098856, 659034566247397820853845450, 1725374928398876195539762836, 4517090274289462986894095759, 11825896005149977207399831924, 30960597962521397519820017125, 81055898325136073121089455887, 212207097898330537381506825687, 555565397140742970099547973823, 1454489097065673235069371003496, 3807901901139826459413032854579, 9969216620520905591778663198626, 26099747988757089213140828020698, 68330027402418759842079563424968, 178890334331835985901969347379982, 468340975819762789041571448969381, 1226132593580799563578230940039894, 3210056805829330266404093252176771, 8404037825720579965055992578546441, 22002056674959187087607772007577757, 57602132206410536215455098492420481, 150804339958779531394133073566154310, 394810887698942277637695222398987101, 1033628323196075740860454794016699784, 2706074082005341823626673560422901404, 7084593923052063487385574688795586390, 18547707687615076153262068109051025432, 48558529140721620001864664844531829067, 127127879736406693911259996836893144008, 332825110072212281849771466490844971441, 871347450487657791873766684285036511379, 2281217241405616374242953149663054049011, 5972304273758901891797941891301704612563, 15635695579930510423036570777437217746868, 40934782466151471621083166947400264548887, 107168651818761588927755723077544207746046, 280571172990608664137269588310793622386414, 734544867154015141434224213905959186812276
Generating function in Maple syntax:
-(3*x^5-15*x^4+24*x^3-19*x^2+7*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^3
Generating function in latex syntax:
-\frac{3 x^{5}-15 x^{4}+24 x^{3}-19 x^{2}+7 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-3*x**5 + 15*x**4 - 24*x**3 + 19*x**2 - 7*x + 1)/((x - 1)**3*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^3*F(x)+3*x^5-15*x^4+24*x^3-19*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+3 x^{5}-15 x^{4}+24 x^{3}-19 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
1/10*(3*5^(1/2)+5)*(3/2-1/2*5^(1/2))^(-n)+1/10*(-3*5^(1/2)+5)*(3/2+1/2*5^(1/2))^(-n)+1/2*n^2+1/2*n-3*2^n+3
Explicit closed form in latex syntax:
\frac{\left(3 \sqrt{5}+5\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-3 \sqrt{5}+5\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{n^{2}}{2}+\frac{n}{2}-3 \,2^{n}+3
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(n+3) = 1/2*n^2+2*a(n)-7*a(n+1)+5*a(n+2)+1/2*n+1, n >= 6
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) = 66
a \! \left(n +3\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{n}{2}+1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15748/
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[17,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[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] = x
F[13,x] = F[1,x]+F[10,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[9,x]
F[17,x] = F[18,x]+F[24,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[21,x]
F[24,x] = F[25,x]+F[26,x]+F[57,x]
F[25,x] = 0
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[40,x]
F[28,x] = F[29,x]+F[6,x]
F[29,x] = F[25,x]+F[30,x]+F[31,x]
F[30,x] = F[12,x]*F[28,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[9,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[10,x]+F[34,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[33,x]
F[40,x] = F[24,x]+F[41,x]
F[41,x] = 2*F[25,x]+F[42,x]+F[43,x]
F[42,x] = F[12,x]*F[40,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[49,x]
F[45,x] = F[22,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[45,x]
F[49,x] = F[50,x]+F[54,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[22,x]+F[53,x]
F[53,x] = F[51,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[49,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[17,x]+F[59,x]
F[59,x] = F[60,x]+F[63,x]
F[60,x] = F[23,x]+F[25,x]+F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[36,x]+F[52,x]
F[63,x] = 2*F[25,x]+F[64,x]+F[80,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[66,x]+F[71,x]
F[66,x] = F[14,x]+F[67,x]
F[67,x] = 2*F[25,x]+F[68,x]+F[69,x]
F[68,x] = F[12,x]*F[66,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[33,x]
F[71,x] = F[46,x]+F[72,x]
F[72,x] = 3*F[25,x]+F[73,x]+F[74,x]
F[73,x] = F[12,x]*F[71,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[53,x]+F[77,x]
F[77,x] = F[78,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[76,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[59,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_{17}\! \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_{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) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \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_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{57}\! \left(x \right)
F_{25}\! \left(x \right) = 0
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_{40}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{6}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{9}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)
F_{40}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{22}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{22}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{49}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{17}\! \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_{23}\! \left(x \right)+F_{25}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{36}\! \left(x \right)+F_{52}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{64}\! \left(x \right)+F_{80}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{71}\! \left(x \right)
F_{66}\! \left(x \right) = F_{14}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{33}\! \left(x \right)
F_{71}\! \left(x \right) = F_{46}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 3 F_{25}\! \left(x \right)+F_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{53}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{76}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{59}\! \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_17(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_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), x)
Eq(F_13(x), F_1(x) + F_10(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_9(x))
Eq(F_17(x), F_18(x) + F_24(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_21(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_57(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_40(x))
Eq(F_28(x), F_29(x) + F_6(x))
Eq(F_29(x), F_25(x) + F_30(x) + F_31(x))
Eq(F_30(x), F_12(x)*F_28(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_9(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_10(x) + F_34(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_33(x))
Eq(F_40(x), F_24(x) + F_41(x))
Eq(F_41(x), 2*F_25(x) + F_42(x) + F_43(x))
Eq(F_42(x), F_12(x)*F_40(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_49(x))
Eq(F_45(x), F_22(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_45(x))
Eq(F_49(x), F_50(x) + F_54(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_22(x) + F_53(x))
Eq(F_53(x), F_51(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_49(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_17(x) + F_59(x))
Eq(F_59(x), F_60(x) + F_63(x))
Eq(F_60(x), F_23(x) + F_25(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_36(x) + F_52(x))
Eq(F_63(x), 2*F_25(x) + F_64(x) + F_80(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_71(x))
Eq(F_66(x), F_14(x) + F_67(x))
Eq(F_67(x), 2*F_25(x) + F_68(x) + F_69(x))
Eq(F_68(x), F_12(x)*F_66(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_33(x))
Eq(F_71(x), F_46(x) + F_72(x))
Eq(F_72(x), 3*F_25(x) + F_73(x) + F_74(x))
Eq(F_73(x), F_12(x)*F_71(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_53(x) + F_77(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_76(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_59(x))
Pack JSON:
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Specification JSON:
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