0123_0321_1032_1230_3021
Counting sequence:
1, 1, 2, 6, 19, 46, 99, 223, 495, 1087, 2367, 5119, 11007, 23551, 50175, 106495, 225279, 475135, 999423, 2097151, 4390911, 9175039, 19136511, 39845887, 82837503, 171966463, 356515839, 738197503, 1526726655, 3154116607, 6509559807, 13421772799, 27648851967, 56908316671, 117037858815, 240518168575, 493921239039, 1013612281855, 2078764171263, 4260607557631, 8727373545471, 17867063951359, 36558761623551, 74766790688767, 152832116260863, 312261302288383, 637716744110079, 1301821767286783, 2656420092706815, 5418393301680127, 11047892835893247, 22517998136852479, 45880421203836927, 93449692267937791, 190277084256403455, 387309567953862655, 788129934789836799, 1603281467343896575, 3260606130216239103, 6629298651489370111, 13474770085092524031, 27381885734412615679, 55628462597280366591, 112986307451471003647, 229431379416762548223, 465780287861166178303, 945395633777614520319, 1918461383665793368063, 3892262999552715390975, 7895206463547688091647, 16011773855979890802687, 32466269569728810844159, 65817982854995680165887, 133406853141067477286911, 270355481144287188484095, 547794512012878844788735, 1109756123474366625218559, 2247846445845951121719295, 4552361289486337986002943, 9218059374561547457134591, 18662792340300837884526591, 37778931862957161709567999, 76464558090625295300165631, 154742504910672534362390527, 313111787280188956248899583, 633477129478065687546036223, 1281461368791506925188546559, 2591936957253764950570041343, 5241902353849032101525979135, 10599861586381068603823751167, 21431836930128146009191088127, 43327901374988309621469347839, 87584257779440654449113038847, 177025425617809379310574764031, 357764671353474899445846900735, 722956982942662080541088546815, 1460769246356748724380966584319, 2951249053656346575359512150015, 5961919229198391403914182262783, 12042680702168179314218680451071, 24323045891879151641217992753151
Generating function in Maple syntax:
(12*x^8-20*x^7+3*x^6+9*x^5-x^4-5*x^2+4*x-1)/(x-1)/(2*x-1)^2
Generating function in latex syntax:
\frac{12 x^{8}-20 x^{7}+3 x^{6}+9 x^{5}-x^{4}-5 x^{2}+4 x -1}{\left(x -1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(12*x**8 - 20*x**7 + 3*x**6 + 9*x**5 - x**4 - 5*x**2 + 4*x - 1)/((x - 1)*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)^2*F(x)+12*x^8-20*x^7+3*x^6+9*x^5-x^4-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+12 x^{8}-20 x^{7}+3 x^{6}+9 x^{5}-x^{4}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,n = 4,19,n = 5,46,-1+1/16*(3*n+7)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 19 & n =4 \\ 46 & n =5 \\ -1+\frac{\left(3 n +7\right) 2^{n}}{16} & \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) = 46
a(6) = 99
a(7) = 223
a(8) = 495
a(n+2) = -4*a(n)+4*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) = 46
a \! \left(6\right) = 99
a \! \left(7\right) = 223
a \! \left(8\right) = 495
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-1, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17713/
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[28,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[4,x]
F[21,x] = F[22,x]
F[22,x] = x^2
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[15,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[4,x]
F[28,x] = F[29,x]+F[59,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[1,x]+F[15,x]
F[33,x] = F[29,x]+F[34,x]
F[34,x] = F[18,x]+F[35,x]+F[57,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[39,x]
F[37,x] = F[15,x]+F[38,x]
F[38,x] = F[26,x]
F[39,x] = F[34,x]+F[40,x]
F[40,x] = 2*F[18,x]+F[41,x]+F[45,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[38,x]
F[44,x] = F[40,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[4,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[25,x]+F[4,x]
F[51,x] = F[47,x]+F[52,x]
F[52,x] = 2*F[18,x]+F[45,x]+F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[25,x]
F[56,x] = F[52,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[29,x]+F[47,x]
F[59,x] = F[18,x]+F[60,x]+F[77,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[65,x]
F[62,x] = F[15,x]+F[63,x]
F[63,x] = F[18,x]+F[19,x]+F[64,x]
F[64,x] = F[24,x]*F[4,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[18,x]+F[57,x]+F[60,x]
F[67,x] = F[18,x]+F[68,x]+F[73,x]+F[75,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[38,x]
F[71,x] = F[72,x]
F[72,x] = 2*F[18,x]+F[45,x]+F[68,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[47,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[34,x]+F[52,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[81,x]
F[79,x] = F[29,x]+F[80,x]
F[80,x] = F[48,x]
F[81,x] = F[34,x]+F[82,x]
F[82,x] = 2*F[18,x]+F[53,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_{28}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = x^{2}
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{59}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{57}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \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_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{25}\! \left(x \right)+F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{18}\! \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_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{25}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{29}\! \left(x \right)+F_{47}\! \left(x \right)
F_{59}\! \left(x \right) = F_{18}\! \left(x \right)+F_{60}\! \left(x \right)+F_{77}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{67}\! \left(x \right) = F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{38}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{68}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{47}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{34}\! \left(x \right)+F_{52}\! \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_{29}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{48}\! \left(x \right)
F_{81}\! \left(x \right) = F_{34}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{53}\! \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_28(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_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), x**2)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_15(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_4(x))
Eq(F_28(x), F_29(x) + F_59(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_1(x) + F_15(x))
Eq(F_33(x), F_29(x) + F_34(x))
Eq(F_34(x), F_18(x) + F_35(x) + F_57(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_39(x))
Eq(F_37(x), F_15(x) + F_38(x))
Eq(F_38(x), F_26(x))
Eq(F_39(x), F_34(x) + F_40(x))
Eq(F_40(x), 2*F_18(x) + F_41(x) + F_45(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_38(x))
Eq(F_44(x), F_40(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_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_25(x) + F_4(x))
Eq(F_51(x), F_47(x) + F_52(x))
Eq(F_52(x), 2*F_18(x) + F_45(x) + F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_25(x))
Eq(F_56(x), F_52(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_29(x) + F_47(x))
Eq(F_59(x), F_18(x) + F_60(x) + F_77(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_65(x))
Eq(F_62(x), F_15(x) + F_63(x))
Eq(F_63(x), F_18(x) + F_19(x) + F_64(x))
Eq(F_64(x), F_24(x)*F_4(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_18(x) + F_57(x) + F_60(x))
Eq(F_67(x), F_18(x) + F_68(x) + F_73(x) + F_75(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_38(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), 2*F_18(x) + F_45(x) + F_68(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_47(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_34(x) + F_52(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_29(x) + F_80(x))
Eq(F_80(x), F_48(x))
Eq(F_81(x), F_34(x) + F_82(x))
Eq(F_82(x), 2*F_18(x) + F_53(x) + F_73(x))
Pack JSON:
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Specification JSON:
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