0123_0132_1023_1230_1320

Counting sequence:
1, 1, 2, 6, 19, 56, 149, 367, 859, 1943, 4294, 9337, 20068, 42764, 90538, 190716, 400121, 836694, 1744831, 3630229, 7537893, 15624733, 32337848, 66837151, 137972534, 284501706, 586052364, 1206098682, 2480017219, 5095402388, 10461101129, 21462083947, 44002781023, 90160927379, 184629574618, 377869717621, 772952690296, 1580319138104, 3229445157406, 6596470691304, 13468048116413, 27486222295938, 56072555295043, 114345103169521, 233089821248649, 474978273241081, 967552838645804, 1970296693220347, 4010972880576554, 8162700643305558, 16606904407076064, 33776804305123782, 68679582198393703, 139611083429325344, 283725959386176173, 576459430146099751, 1170933763820841571, 2377897141798812239, 4827853199792876206, 9799823726957095729, 19887881291518712908, 40352228936089144868, 81857388438986240338, 166020634550135572116, 336652978843849031489, 682529368112652733806, 1383505542412265411719, 2803904673473300212237, 5681596485856039306989, 11510767187417705986453, 23316682705745316432224, 47223661910695841106967, 95627916556686147739358, 193617018158230674896034, 391956405717331606035828, 793357549121826670337106, 1605604571814556700394571, 3248988087852919509199724, 6573534059432027067382481, 13298183878677005453865379, 26898599264619064599230503, 54401661523767962855896523, 110012249004234470354401858, 222442349869504633596198061, 449720403376358133897102304, 909112213890330085735714256, 1837567241834081372816465734, 3713820111416114798317124352, 7505011477747436917458774149, 15164765464385701342016860698, 30639015945032773779138750859, 61897001920128418694844230377, 125031943896402424690084996113, 252539767898655997954582814449, 510031295998594112058874106708, 1029966112382892249391667757811, 2079739265509912161306560030546, 4199092612463939052609457109934, 8477413387744686582136861766856, 17113283101007428539984780100254, 34543478852863985270192108259823

Generating function in Maple syntax:
-(4*x^9-4*x^8-6*x^7+9*x^6-7*x^5-6*x^4+22*x^3-19*x^2+7*x-1)/(x^2+x-1)/(2*x-1)^2/(-1+x)^3

Generating function in latex syntax:
-\frac{4 x^{9}-4 x^{8}-6 x^{7}+9 x^{6}-7 x^{5}-6 x^{4}+22 x^{3}-19 x^{2}+7 x -1}{\left(x^{2}+x -1\right) \left(2 x -1\right)^{2} \left(-1+x \right)^{3}}

Generating function in sympy syntax:
(-4*x**9 + 4*x**8 + 6*x**7 - 9*x**6 + 7*x**5 + 6*x**4 - 22*x**3 + 19*x**2 - 7*x + 1)/((x - 1)**3*(2*x - 1)**2*(x**2 + x - 1))

Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(2*x-1)^2*(-1+x)^3*F(x)+4*x^9-4*x^8-6*x^7+9*x^6-7*x^5-6*x^4+22*x^3-19*x^2+7*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(2 x -1\right)^{2} \left(-1+x \right)^{3} F \! \left(x \right)+4 x^{9}-4 x^{8}-6 x^{7}+9 x^{6}-7 x^{5}-6 x^{4}+22 x^{3}-19 x^{2}+7 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/4*(4*5^(1/2)-8)*(-1/2*5^(1/2)-1/2)^(-n)+1/4*(-4*5^(1/2)-8)*(1/2*5^(1/2)-1/2)^(-n)+1/4*(n+9)*2^n-1/2*n^2-1/2*n+6)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(4 \sqrt{5}-8\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{4}+\frac{\left(-4 \sqrt{5}-8\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{4}+\frac{\left(n +9\right) 2^{n}}{4}-\\\frac{n^{2}}{2}-\frac{n}{2}+6 & \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) = 56
a(6) = 149
a(7) = 367
a(8) = 859
a(9) = 1943
a(n) = 7/4*a(n+2)-5/4*a(n+3)+1/4*a(n+4)-1/8*(n+2)*(n-7), n >= 10

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) = 56
a \! \left(6\right) = 149
a \! \left(7\right) = 367
a \! \left(8\right) = 859
a \! \left(9\right) = 1943
a \! \left(n \right) = \frac{7 a \! \left(n +2\right)}{4}-\frac{5 a \! \left(n +3\right)}{4}+\frac{a \! \left(n +4\right)}{4}-\frac{\left(n +2\right) \left(n -7\right)}{8}, \quad n \geq 10

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16305/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[80,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[35,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[1,x]+F[24,x]
F[27,x] = F[13,x]+F[14,x]+F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[7,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[16,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[31,x]
F[35,x] = F[36,x]+F[54,x]
F[36,x] = F[13,x]+F[21,x]+F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[24,x]+F[30,x]
F[43,x] = F[39,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[34,x]+F[4,x]
F[52,x] = F[48,x]+F[53,x]
F[53,x] = F[45,x]
F[54,x] = F[13,x]+F[55,x]+F[68,x]+F[77,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[64,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[24,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[30,x]
F[64,x] = F[65,x]
F[65,x] = 2*F[13,x]+F[55,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[39,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[47,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[74,x]+F[76,x]
F[74,x] = F[75,x]
F[75,x] = F[24,x]*F[4,x]
F[76,x] = F[72,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[40,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[2,x]+F[48,x]
F[83,x] = F[71,x]+F[79,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{80}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{35}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{54}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \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_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \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_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{34}\! \left(x \right)+F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{45}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right)+F_{55}\! \left(x \right)+F_{68}\! \left(x \right)+F_{77}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{64}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{24}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{30}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{55}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{39}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{47}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{76}\! \left(x \right) = F_{72}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{40}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{2}\! \left(x \right)+F_{48}\! \left(x \right)
F_{83}\! \left(x \right) = F_{71}\! \left(x \right)+F_{79}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_80(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_35(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_1(x) + F_24(x))
Eq(F_27(x), F_13(x) + F_14(x) + F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_7(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_16(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_31(x))
Eq(F_35(x), F_36(x) + F_54(x))
Eq(F_36(x), F_13(x) + F_21(x) + F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_24(x) + F_30(x))
Eq(F_43(x), F_39(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_34(x) + F_4(x))
Eq(F_52(x), F_48(x) + F_53(x))
Eq(F_53(x), F_45(x))
Eq(F_54(x), F_13(x) + F_55(x) + F_68(x) + F_77(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_64(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_24(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_30(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), 2*F_13(x) + F_55(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_39(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_47(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_76(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_24(x)*F_4(x))
Eq(F_76(x), F_72(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_40(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_2(x) + F_48(x))
Eq(F_83(x), F_71(x) + F_79(x))
Pack JSON:
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Specification JSON:
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