0123_0231_1032_2103_3012

Counting sequence:
1, 1, 2, 6, 19, 47, 109, 263, 636, 1536, 3709, 8955, 21620, 52196, 126013, 304223, 734460, 1773144, 4280749, 10334643, 24950036, 60234716, 145419469, 351073655, 847566780, 2046207216, 4939981213, 11926169643, 28792320500, 69510810644, 167813941789, 405138694223, 978091330236, 2361321354696, 5700734039629, 13762789433955, 33226312907540, 80215415249036, 193657143405613, 467529702060263, 1128716547526140, 2724962797112544, 6578642141751229, 15882247080615003, 38343136302981236, 92568519686577476, 223480175676136189, 539528871038849855, 1302537917753835900, 3144604706546521656, 7591747330846879213, 18328099368240280083, 44247946067327439380, 106823991502895158844, 257895929073117757069, 622615849649130672983, 1503127628371379103036, 3628871106391888879056, 8760869841155156861149, 21150610788702202601355, 51062091418559562063860, 123274793625821326729076, 297611678670202215522013, 718498150966225757773103, 1734607980602653731068220, 4187714112171533219909544, 10110036204945720170887309, 24407786522062973561684163, 58925609249071667294255636, 142259005020206308150195436, 343443619289484283594646509, 829146243599174875339488455, 2001736106487834034273623420, 4832618456574842943886735296, 11666973019637519922047094013, 28166564495849882787980923323, 68000102011337285498008940660, 164166768518524453783998804644, 396333639048386193066006549949, 956834046615296839916011904543, 2310001732278979872898030359036, 5576837511173256585712072622616, 13463676754625493044322175604269, 32504191020424242674356423831155, 78472058795473978393035023266580, 189448308611372199460426470364316, 457368676018218377313887963995213, 1104185660647808954088202398354743, 2665739997313836285490292760704700, 6435665655275481525068787919764144, 15537071307864799335627868600232989, 37509808271005080196324525120230123, 90556687849874959728276918840693236, 218623183970754999652878362801616596, 527803055791384959034033644443926429, 1274229295553524917720945651689469455, 3076261646898434794475924947822865340, 7426752589350394506672795547335200136, 17929766825599223807821516042493265613, 43286286240548842122315827632321731363, 104502339306696908052453171307136728340

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

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

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

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

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

Explicit closed form in Maple syntax:
piecewise(n = 0,-59,n = 1,25,n = 2,-11,n = 3,3,n = 4,-1,n = 5,3,0)-1/2+21*(-1-2^(1/2))^(-n)*2^(1/2)-21*(2^(1/2)-1)^(-n)*2^(1/2)+121/4*(-1-2^(1/2))^(-n)+121/4*(2^(1/2)-1)^(-n)

Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}-59 & n =0 \\ 25 & n =1 \\ -11 & n =2 \\ 3 & n =3 \\ -1 & n =4 \\ 3 & n =5 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)-\frac{1}{2}+21 \left(-1-\sqrt{2}\right)^{-n} \sqrt{2}-21 \left(\sqrt{2}-1\right)^{-n} \sqrt{2}+\frac{121 \left(-1-\sqrt{2}\right)^{-n}}{4}+\frac{121 \left(\sqrt{2}-1\right)^{-n}}{4}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 47
a(6) = 109
a(7) = 263
a(8) = 636
a(n+2) = a(n)+2*a(n+1)+1, 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) = 47
a \! \left(6\right) = 109
a \! \left(7\right) = 263
a \! \left(8\right) = 636
a \! \left(n +2\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+1, \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17271/
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[18,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[4,x]
F[18,x] = F[19,x]+F[36,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[10,x]+F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[35,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[28,x]+F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[11,x]*F[4,x]
F[30,x] = F[31,x]+F[33,x]
F[31,x] = F[32,x]
F[32,x] = F[19,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[23,x]*F[4,x]
F[35,x] = F[22,x]*F[4,x]
F[36,x] = F[24,x]+F[37,x]+F[58,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[48,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[1,x]+F[4,x]
F[43,x] = F[24,x]+F[44,x]+F[45,x]
F[44,x] = F[17,x]*F[4,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[11,x]+F[47,x]
F[47,x] = F[44,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[19,x]+F[31,x]
F[52,x] = 2*F[24,x]+F[53,x]+F[55,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[31,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[23,x]+F[57,x]
F[57,x] = F[53,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[68,x]
F[60,x] = F[19,x]+F[61,x]
F[61,x] = F[24,x]+F[35,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[66,x]
F[64,x] = F[4,x]+F[65,x]
F[65,x] = F[24,x]+F[29,x]+F[44,x]
F[66,x] = F[31,x]+F[67,x]
F[67,x] = 2*F[24,x]+F[34,x]+F[53,x]
F[68,x] = F[69,x]+F[76,x]
F[69,x] = F[24,x]+F[25,x]+F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[22,x]+F[72,x]
F[72,x] = F[69,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[54,x]
F[76,x] = 2*F[24,x]+F[74,x]+F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[81,x]
F[79,x] = F[80,x]
F[80,x] = x^2
F[81,x] = F[82,x]
F[82,x] = F[31,x]*F[4,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_{18}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{36}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)+F_{58}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{48}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{24}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{44}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)+F_{31}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{31}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{23}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{68}\! \left(x \right)
F_{60}\! \left(x \right) = F_{19}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{24}\! \left(x \right)+F_{35}\! \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_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)+F_{44}\! \left(x \right)
F_{66}\! \left(x \right) = F_{31}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{34}\! \left(x \right)+F_{53}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{76}\! \left(x \right)
F_{69}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{22}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{69}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{54}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{74}\! \left(x \right)+F_{77}\! \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_{81}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = x^{2}
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \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_18(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_19(x) + F_36(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_10(x) + F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_35(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_28(x) + F_4(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_11(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_33(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_19(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_23(x)*F_4(x))
Eq(F_35(x), F_22(x)*F_4(x))
Eq(F_36(x), F_24(x) + F_37(x) + F_58(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_48(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_1(x) + F_4(x))
Eq(F_43(x), F_24(x) + F_44(x) + F_45(x))
Eq(F_44(x), F_17(x)*F_4(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_11(x) + F_47(x))
Eq(F_47(x), F_44(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_19(x) + F_31(x))
Eq(F_52(x), 2*F_24(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_31(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_23(x) + F_57(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_68(x))
Eq(F_60(x), F_19(x) + F_61(x))
Eq(F_61(x), F_24(x) + F_35(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_66(x))
Eq(F_64(x), F_4(x) + F_65(x))
Eq(F_65(x), F_24(x) + F_29(x) + F_44(x))
Eq(F_66(x), F_31(x) + F_67(x))
Eq(F_67(x), 2*F_24(x) + F_34(x) + F_53(x))
Eq(F_68(x), F_69(x) + F_76(x))
Eq(F_69(x), F_24(x) + F_25(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_22(x) + F_72(x))
Eq(F_72(x), F_69(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_54(x))
Eq(F_76(x), 2*F_24(x) + F_74(x) + F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_81(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), x**2)
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_31(x)*F_4(x))
Pack JSON:
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Specification JSON:
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