0213_0231_1032_1203_3012
Counting sequence:
1, 1, 2, 6, 19, 52, 134, 341, 872, 2248, 5833, 15198, 39692, 103791, 271574, 710802, 1860679, 4871056, 12752282, 33385553, 87404108, 228826468, 599074957, 1568398026, 4106118704, 10749957627, 28143753674, 73681302846, 192900154267, 505019159308, 1322157322958, 3461452808813, 9062201102672, 23725150498336, 62113250391409, 162614600674902, 425730551632244, 1114577054220711, 2918000611028702, 7639424778864138, 20000273725562383, 52361396397821608, 137083915467900962, 358890350005879721, 939587134549736564, 2459871053643328252, 6440026026380246389, 16860207025497409026, 44140595050111978712, 115561578124838525043, 302544139324403594258, 792070839848372255478, 2073668380220713169827, 5428934300813767251556, 14213134522220588582294, 37210469265847998492677, 97418273275323406892984, 255044350560122222183416, 667714778405043259654297, 1748099984655007556776398, 4576585175559979410671708, 11981655542024930675235423, 31368381450514812615031142, 82123488809519507169854466, 215002084978043708894528599, 562882766124611619513727552, 1473646213395791149646650154, 3858055874062761829426218881, 10100521408792494338632002332, 26443508352314721186469783828, 69230003648151669220777344733, 181246502592140286475862245818, 474509504128269190206809388032, 1242282009792667284144565913451, 3252336525249732662226888347354, 8514727565956530702536099123502, 22291846172619859445381409017899, 58360810951903047633608127924796, 152790586683089283455442974750942, 400010949097364802732720796322333, 1047242260609005124742719414210208, 2741715832729650571495437446302288, 7177905237579946589743592924690497, 18791999880010189197735341327762886, 49198094402450621003462431058591684, 128802283327341673812651951848005527, 337208755579574400434493424485418094, 882823983411381527490828321608241786, 2311263194654570182037991540339300127, 6050965600552329018623146299409651288, 15841633607002416873831447357889646258, 41473935220454921602871195774259279833, 108580172054362347934782139964888185412, 284266580942632122201475224120405268396, 744219570773534018669643532396327611589, 1948392131377969933807455373068577558002, 5100956823360375782752722586809405053864, 13354478338703157414450712387359637594851, 34962478192749096460599414575269507721762, 91532956239544131967347531338448885561318, 239636390525883299441443179440077148952883
Generating function in Maple syntax:
-(2*x-1)*(x^5-2*x^4-3*x^2+3*x-1)/(x^2-3*x+1)/(x-1)^3
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x^{5}-2 x^{4}-3 x^{2}+3 x -1\right)}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(1 - 2*x)*(x**5 - 2*x**4 - 3*x**2 + 3*x - 1)/((x - 1)**3*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^3*F(x)+(2*x-1)*(x^5-2*x^4-3*x^2+3*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(2 x -1\right) \left(x^{5}-2 x^{4}-3 x^{2}+3 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n^2+5-5*n+(3/2-1/2*5^(1/2))^(1-n)+(3/2+1/2*5^(1/2))^(1-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ n^{2}+5-5 n +\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n} & \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) = 134
a(n+2) = -n^2-a(n)+3*a(n+1)+3*n+1, n >= 7
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) = 134
a \! \left(n +2\right) = -n^{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+3 n +1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20304/
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[17,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[6,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[4,x]
F[12,x] = F[13,x]+F[16,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[35,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,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[24,x]+F[34,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]+F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[14,x]*F[4,x]
F[29,x] = F[30,x]+F[32,x]
F[30,x] = F[23,x]+F[24,x]+F[31,x]
F[31,x] = F[18,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[22,x]*F[4,x]
F[34,x] = F[21,x]*F[4,x]
F[35,x] = F[23,x]+F[36,x]+F[64,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[47,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[1,x]+F[39,x]
F[42,x] = F[23,x]+F[43,x]+F[45,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[39,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[14,x]
F[47,x] = F[48,x]+F[52,x]
F[48,x] = F[23,x]+F[36,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[18,x]+F[51,x]
F[51,x] = F[49,x]
F[52,x] = F[23,x]+F[53,x]+F[60,x]+F[62,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[56,x]
F[56,x] = F[39,x]*F[4,x]
F[57,x] = F[58,x]
F[58,x] = 2*F[23,x]+F[53,x]+F[59,x]
F[59,x] = F[4,x]*F[51,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[51,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[22,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[77,x]
F[66,x] = F[18,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[66,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[76,x]
F[73,x] = F[18,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[4,x]*F[73,x]
F[76,x] = F[71,x]
F[77,x] = F[78,x]
F[78,x] = F[23,x]+F[24,x]+F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[73,x]+F[81,x]
F[81,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{17}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{35}\! \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_{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_{24}\! \left(x \right)+F_{34}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)+F_{64}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{52}\! \left(x \right)
F_{48}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{23}\! \left(x \right)+F_{53}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{53}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{51}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{22}\! \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_{77}\! \left(x \right)
F_{66}\! \left(x \right) = F_{18}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{66}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{71}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{73}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_17(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_6(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_4(x))
Eq(F_12(x), F_13(x) + F_16(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_35(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(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) + F_24(x) + F_34(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x) + F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_14(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_32(x))
Eq(F_30(x), F_23(x) + F_24(x) + F_31(x))
Eq(F_31(x), F_18(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_22(x)*F_4(x))
Eq(F_34(x), F_21(x)*F_4(x))
Eq(F_35(x), F_23(x) + F_36(x) + F_64(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_47(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_1(x) + F_39(x))
Eq(F_42(x), F_23(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_39(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_14(x))
Eq(F_47(x), F_48(x) + F_52(x))
Eq(F_48(x), F_23(x) + F_36(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_18(x) + F_51(x))
Eq(F_51(x), F_49(x))
Eq(F_52(x), F_23(x) + F_53(x) + F_60(x) + F_62(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_39(x)*F_4(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), 2*F_23(x) + F_53(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_51(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_51(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_22(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_77(x))
Eq(F_66(x), F_18(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_66(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_76(x))
Eq(F_73(x), F_18(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_4(x)*F_73(x))
Eq(F_76(x), F_71(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_23(x) + F_24(x) + F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_73(x) + F_81(x))
Eq(F_81(x), F_78(x))
Pack JSON:
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Specification JSON:
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