0123_0132_0321_1032_1230

Counting sequence:
1, 1, 2, 6, 19, 54, 149, 405, 1087, 2888, 7609, 19907, 51771, 133946, 345005, 885137, 2262983, 5767692, 14659249, 37164543, 94005923, 237289918, 597830405, 1503553549, 3775387791, 9465832528, 23700509929, 59265269947, 148021342027, 369286955970, 920343550109, 2291449650441, 5699942335639, 14166208885268, 35178808717921, 87291598108151, 216443997120051, 536311348506950, 1328012198637173, 3286358510946437, 8127720255363551, 20089803856505688, 49630328672872729, 122544467446078899, 302430276757183835, 746021053588580938, 1839415462382767757, 4533354167879094913, 11168071255639336551, 27501893783679504412, 67698600489540197137, 166584975200365338607, 409767053432023605955, 1007601967585523453646, 2476835262187045051109, 6086483924648673534717, 14952090250446486616111, 36720450136154895736928, 90154849082944870525513, 221283651133035070629035, 542991015571184471901291, 1332056913550733368508114, 3266956169445479377188221, 8010463137262677813502457, 19636721540385634553699639, 48126078295684531710532644, 117922061383470667103533249, 288878739643708388964766695, 707529801084768460256170067, 1732547401222089862970481238, 4241692982760518308406984789, 10382679184615111277697529717, 25409611366966280581838054271, 62173767766370736675358737256, 152103438610329422118561736825, 372045094256095981518479725987, 909868817371275854553522427003, 2224797558765223030027524571482, 5439168784683627062810572791149, 13295559657462862791454672587633, 32494842008052029065533924055303, 79406776019782659397956535309868, 194016012648491501232129029987185, 473975570864729707079012680520159, 1157748312074753159194432596811491, 2827571279955804558293232370947038, 6904870199566301585235884538097349, 16859369419189854880500330342729197, 41159703845728844959162190214122447, 100473024466314659176411329647694960, 245230342297475225680083732253520809, 598477135455166349650363178519474331, 1460396055514727465576477740765953419, 3563235557492361601108438347363086818, 8693011234821850848999261461589022301, 21205512466788603981823895010046627561, 51722689112026540359286825986790164951, 126144451017749188476394029733348228660, 307617364744967406410707625458037052193, 750084288029476483271071243408244464087, 1828801929444947825998006777796720672371

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

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

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

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

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

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,1/16*((6*n-53)*2^(1/2)+10*n-72)*(-1-2^(1/2))^(-n)-3/8*(2^(1/2)-1)^(-n)*((n-53/6)*2^(1/2)-5/3*n+12))

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(\left(6 n -53\right) \sqrt{2}+10 n -72\right) \left(-1-\sqrt{2}\right)^{-n}}{16}-\\\frac{3 \left(\sqrt{2}-1\right)^{-n} \left(\left(n -\frac{53}{6}\right) \sqrt{2}-\frac{5 n}{3}+12\right)}{8} & \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) = 149
a(7) = 405
a(n+4) = -a(n)-4*a(n+1)-2*a(n+2)+4*a(n+3), n >= 8

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) = 149
a \! \left(7\right) = 405
a \! \left(n +4\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-2 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16222/
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[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[76,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[31,x]
F[23,x] = F[12,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[28,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[14,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[12,x]+F[30,x]
F[30,x] = F[16,x]
F[31,x] = F[32,x]+F[61,x]
F[32,x] = F[15,x]+F[21,x]+F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[17,x]+F[42,x]
F[42,x] = F[30,x]
F[43,x] = F[35,x]+F[44,x]
F[44,x] = 2*F[15,x]+F[45,x]+F[58,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[51,x]
F[47,x] = F[30,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[30,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = 2*F[15,x]+F[45,x]+F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[35,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[52,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[54,x]+F[60,x]
F[60,x] = F[52,x]
F[61,x] = F[15,x]+F[62,x]+F[70,x]+F[74,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[67,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[16,x]
F[66,x] = F[49,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = 2*F[15,x]+F[53,x]+F[62,x]
F[69,x] = F[56,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[54,x]+F[72,x]
F[72,x] = F[73,x]
F[73,x] = 2*F[15,x]+F[45,x]+F[53,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[32,x]+F[52,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[34,x]+F[78,x]
F[78,x] = F[32,x]+F[73,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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{76}\! \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_{31}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{61}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \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_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{4}\! \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_{17}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{30}\! \left(x \right)
F_{43}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{58}\! \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_{51}\! \left(x \right)
F_{47}\! \left(x \right) = F_{30}\! \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_{30}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{35}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{52}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \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_{67}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{16}\! \left(x \right)
F_{66}\! \left(x \right) = F_{49}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{62}\! \left(x \right)
F_{69}\! \left(x \right) = F_{56}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{54}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{32}\! \left(x \right)+F_{52}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{34}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{32}\! \left(x \right)+F_{73}\! \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_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_76(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_31(x))
Eq(F_23(x), F_12(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_14(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_12(x) + F_30(x))
Eq(F_30(x), F_16(x))
Eq(F_31(x), F_32(x) + F_61(x))
Eq(F_32(x), F_15(x) + F_21(x) + F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_17(x) + F_42(x))
Eq(F_42(x), F_30(x))
Eq(F_43(x), F_35(x) + F_44(x))
Eq(F_44(x), 2*F_15(x) + F_45(x) + F_58(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_51(x))
Eq(F_47(x), F_30(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_30(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), 2*F_15(x) + F_45(x) + F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_35(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_52(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_54(x) + F_60(x))
Eq(F_60(x), F_52(x))
Eq(F_61(x), F_15(x) + F_62(x) + F_70(x) + F_74(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_67(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_16(x))
Eq(F_66(x), F_49(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), 2*F_15(x) + F_53(x) + F_62(x))
Eq(F_69(x), F_56(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_54(x) + F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), 2*F_15(x) + F_45(x) + F_53(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_32(x) + F_52(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_34(x) + F_78(x))
Eq(F_78(x), F_32(x) + F_73(x))
Pack JSON:
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Specification JSON:
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