0132_0213_0231_2130_3021

Counting sequence:
1, 1, 2, 6, 19, 54, 141, 357, 902, 2298, 5910, 15310, 39848, 104001, 271849, 711154, 1861121, 4871602, 12752947, 33386353, 87405060, 228827590, 599076268, 1568399546, 4106120454, 10749959629, 28143755951, 73681305422, 192900157167, 505019162558, 1322157326585, 3461452812845, 9062201107138, 23725150503266, 62113250396834, 162614600680854, 425730551638756, 1114577054227817, 2918000611036437, 7639424778872538, 20000273725571485, 52361396397831450, 137083915467911583, 358890350005891161, 939587134549748864, 2459871053643341454, 6440026026380260536, 16860207025497424162, 44140595050111994882, 115561578124838542293, 302544139324403612635, 792070839848372275030, 2073668380220713190603, 5428934300813767273606, 14213134522220588605669, 37210469265847998517429, 97418273275323406919166, 255044350560122222211082, 667714778405043259683502, 1748099984655007556807198, 4576585175559979410704160, 11981655542024930675269585, 31368381450514812615067073, 82123488809519507169892226, 215002084978043708894568249, 562882766124611619513769154, 1473646213395791149646693771, 3858055874062761829426264577, 10100521408792494338632050172, 26443508352314721186469833878, 69230003648151669220777397060, 181246502592140286475862300490, 474509504128269190206809445118, 1242282009792667284144565973021, 3252336525249732662226888409479, 8514727565956530702536099188254, 22291846172619859445381409085351, 58360810951903047633608127995022, 152790586683089283455442974824017, 400010949097364802732720796398333, 1047242260609005124742719414289210, 2741715832729650571495437446384370, 7177905237579946589743592924775738, 18791999880010189197735341327851366, 49198094402450621003462431058683484, 128802283327341673812651951848100729, 337208755579574400434493424485516781, 882823983411381527490828321608344042, 2311263194654570182037991540339406037, 6050965600552329018623146299409760938, 15841633607002416873831447357889759735, 41473935220454921602871195774259397225, 108580172054362347934782139964888306808, 284266580942632122201475224120405393886, 744219570773534018669643532396327741264, 1948392131377969933807455373068577691954, 5100956823360375782752722586809405192186, 13354478338703157414450712387359637737637, 34962478192749096460599414575269507869107, 91532956239544131967347531338448885713318, 239636390525883299441443179440077149109635

Generating function in Maple syntax:
-(x^8-3*x^7+5*x^5-8*x^4+15*x^3-14*x^2+6*x-1)/(x^2-3*x+1)/(-1+x)^4

Generating function in latex syntax:
-\frac{x^{8}-3 x^{7}+5 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(-1+x \right)^{4}}

Generating function in sympy syntax:
(-x**8 + 3*x**7 - 5*x**5 + 8*x**4 - 15*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(-1+x)^4*F(x)+x^8-3*x^7+5*x^5-8*x^4+15*x^3-14*x^2+6*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(-1+x \right)^{4} F \! \left(x \right)+x^{8}-3 x^{7}+5 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/6*n^3+7-25/6*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 \\ 2 & n =2 \\ \frac{n^{3}}{6}+7-\frac{25 n}{6}+\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) = 54
a(6) = 141
a(7) = 357
a(8) = 902
a(n+2) = -1/6*n^3-1/2*n^2-a(n)+3*a(n+1)+14/3*n-2, n >= 9

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) = 141
a \! \left(7\right) = 357
a \! \left(8\right) = 902
a \! \left(n +2\right) = -\frac{n^{3}}{6}-\frac{n^{2}}{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{14 n}{3}-2, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18210/
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[19,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[20,x]+F[30,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[5,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[20,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[31,x]+F[32,x]+F[62,x]
F[31,x] = 0
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[46,x]
F[34,x] = F[35,x]+F[6,x]
F[35,x] = F[31,x]+F[36,x]+F[44,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[10,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[38,x]
F[41,x] = F[17,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[41,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[13,x]
F[46,x] = F[30,x]+F[47,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[51,x]+F[60,x]
F[51,x] = F[31,x]+F[52,x]+F[58,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[56,x]
F[54,x] = F[10,x]+F[55,x]
F[55,x] = F[18,x]+F[31,x]+F[36,x]
F[56,x] = F[51,x]+F[57,x]
F[57,x] = F[48,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[20,x]+F[51,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[50,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[59,x]+F[64,x]
F[64,x] = F[65,x]+F[75,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[14,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[12,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[12,x]*F[72,x]
F[75,x] = 2*F[31,x]+F[76,x]+F[78,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[17,x]
F[78,x] = F[12,x]*F[64,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_{19}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{30}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{5}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \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_{27}\! \left(x \right) = F_{20}\! \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_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{62}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{46}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{6}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{44}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{17}\! \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_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right)
F_{46}\! \left(x \right) = F_{30}\! \left(x \right)+F_{47}\! \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_{51}\! \left(x \right)+F_{60}\! \left(x \right)
F_{51}\! \left(x \right) = F_{31}\! \left(x \right)+F_{52}\! \left(x \right)+F_{58}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{56}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{56}\! \left(x \right) = F_{51}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{48}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{20}\! \left(x \right)+F_{51}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{59}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{75}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{14}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \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_{72}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{17}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \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_19(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_20(x) + F_30(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_5(x))
Eq(F_23(x), F_20(x) + F_24(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))
Eq(F_27(x), F_20(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_31(x) + F_32(x) + F_62(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_46(x))
Eq(F_34(x), F_35(x) + F_6(x))
Eq(F_35(x), F_31(x) + F_36(x) + F_44(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_10(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_38(x))
Eq(F_41(x), F_17(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_41(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_13(x))
Eq(F_46(x), F_30(x) + F_47(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_51(x) + F_60(x))
Eq(F_51(x), F_31(x) + F_52(x) + F_58(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_56(x))
Eq(F_54(x), F_10(x) + F_55(x))
Eq(F_55(x), F_18(x) + F_31(x) + F_36(x))
Eq(F_56(x), F_51(x) + F_57(x))
Eq(F_57(x), F_48(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_20(x) + F_51(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_50(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_59(x) + F_64(x))
Eq(F_64(x), F_65(x) + F_75(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_14(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_12(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_12(x)*F_72(x))
Eq(F_75(x), 2*F_31(x) + F_76(x) + F_78(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_17(x))
Eq(F_78(x), F_12(x)*F_64(x))
Pack JSON:
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Specification JSON:
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