0132_0213_1230_3102
Counting sequence:
1, 1, 2, 6, 20, 61, 161, 384, 855, 1821, 3771, 7677, 15475, 31027, 62044, 123931, 247475, 494218, 987198, 1972423, 3941806, 7879014, 15751132, 31491936, 62968350, 125913216, 251790601, 503526034, 1006966370, 2013798531, 4027385391, 8054434962, 16108334585, 32215812547, 64430250369, 128858289671, 257713017245, 515420288833, 1030831301658, 2061647618093, 4123271016581, 8246502875664, 16492942427812, 32985782435041, 65971399193880, 131942530366996, 263884627121866, 527768552705234, 1055535970364560, 2111070104260566, 4222137237134511, 8444269666554572, 16888531554162620, 33777050521833617, 67554080678414269, 135108128405286036, 270216203493996715, 540432320720114873, 1080864501856035543, 2161728777860279713, 4323457190284878231, 8646913789282612639, 17293826621842754792, 34587651695676277071, 69175300886621261735, 138350597720502439526, 276701188883533975706, 553402367156857470859, 1106804717146034100050, 2213609406514177457426, 4427218768082783949256, 8854437463442106850132, 17708874809215182389330, 35417749428037873167612, 70835498548014224208869, 141670996597574435525526, 283341992388633336929510, 566683983472297127796463, 1133367964833109176416603, 2266735926249763728661158, 4533471846971587755102381, 9066943684998781185005439, 18133887355525228343851197, 36267774687633728337672651, 72535549337378394300557665, 145071098613450997877775181, 290142197127707142658978246, 580284394094913641499681409, 1160568787930131786086237465, 2321137575440067431442884236, 4642275150200243225244953376, 9284550299300398672121496517, 18569100596820817928235861340, 37138201190761568662098376032, 74276402376863090713818574294, 148552804746186067622888033766, 297105609480171974830651297380, 594211218940603675441431348978, 1188422437849266916247869441963, 2376844875646853123640877266928, 4753689751210085103791902674656
Generating function in Maple syntax:
-(x^9-7*x^8+17*x^7-13*x^6+16*x^4-27*x^3+20*x^2-7*x+1)/(2*x-1)/(x^2+x-1)/(-1+x)^5
Generating function in latex syntax:
-\frac{x^{9}-7 x^{8}+17 x^{7}-13 x^{6}+16 x^{4}-27 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(-1+x \right)^{5}}
Generating function in sympy syntax:
(-x**9 + 7*x**8 - 17*x**7 + 13*x**6 - 16*x**4 + 27*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**5*(2*x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2+x-1)*(-1+x)^5*F(x)+x^9-7*x^8+17*x^7-13*x^6+16*x^4-27*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}+x -1\right) \left(-1+x \right)^{5} F \! \left(x \right)+x^{9}-7 x^{8}+17 x^{7}-13 x^{6}+16 x^{4}-27 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/24*n^4-5/12*n^3-25/24*n^2-4-19/12*n-1/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+1/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+15/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{n^{4}}{24}-\frac{5 n^{3}}{12}-\frac{25 n^{2}}{24}-4-\frac{19 n}{12}-\frac{\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\frac{\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\\\frac{15 \,2^{n}}{4} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 61
a(6) = 161
a(7) = 384
a(8) = 855
a(9) = 1821
a(n+3) = -2*a(n)-a(n+1)+3*a(n+2)+1/24*(n+2)*(n^3-20*n^2+63*n+12), 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) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 161
a \! \left(7\right) = 384
a \! \left(8\right) = 855
a \! \left(9\right) = 1821
a \! \left(n +3\right) = -2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{\left(n +2\right) \left(n^{3}-20 n^{2}+63 n +12\right)}{24}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15282/
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[17,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[7,x]
F[17,x] = F[18,x]+F[27,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[6,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[18,x]
F[26,x] = F[22,x]
F[27,x] = F[28,x]+F[29,x]+F[65,x]
F[28,x] = 0
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[42,x]
F[31,x] = F[32,x]+F[7,x]
F[32,x] = F[28,x]+F[33,x]+F[39,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[11,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,x]*F[4,x]
F[38,x] = F[14,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[16,x]+F[41,x]
F[41,x] = F[32,x]
F[42,x] = F[27,x]+F[43,x]
F[43,x] = 2*F[28,x]+F[44,x]+F[57,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[53,x]
F[49,x] = F[11,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[35,x]
F[53,x] = F[46,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[46,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[22,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[22,x]
F[64,x] = F[61,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[18,x]+F[46,x]
F[68,x] = F[69,x]+F[80,x]
F[69,x] = F[28,x]+F[70,x]+F[77,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[31,x]+F[72,x]
F[72,x] = F[22,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[63,x]+F[76,x]
F[76,x] = F[73,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[25,x]+F[79,x]
F[79,x] = F[69,x]
F[80,x] = F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[14,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_{17}\! \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_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
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_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{65}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{42}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{32}\! \left(x \right)
F_{42}\! \left(x \right) = F_{27}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{28}\! \left(x \right)+F_{44}\! \left(x \right)+F_{57}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{53}\! \left(x \right)
F_{49}\! \left(x \right) = F_{11}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{35}\! \left(x \right)
F_{53}\! \left(x \right) = F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{46}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{22}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{22}\! \left(x \right)
F_{64}\! \left(x \right) = F_{61}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{18}\! \left(x \right)+F_{46}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{80}\! \left(x \right)
F_{69}\! \left(x \right) = F_{28}\! \left(x \right)+F_{70}\! \left(x \right)+F_{77}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{31}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{22}\! \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_{63}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{73}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{25}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{69}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{14}\! \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_17(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_7(x))
Eq(F_17(x), F_18(x) + F_27(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_6(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_18(x))
Eq(F_26(x), F_22(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_65(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_42(x))
Eq(F_31(x), F_32(x) + F_7(x))
Eq(F_32(x), F_28(x) + F_33(x) + F_39(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_11(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(x)*F_4(x))
Eq(F_38(x), F_14(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_16(x) + F_41(x))
Eq(F_41(x), F_32(x))
Eq(F_42(x), F_27(x) + F_43(x))
Eq(F_43(x), 2*F_28(x) + F_44(x) + F_57(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_53(x))
Eq(F_49(x), F_11(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_35(x))
Eq(F_53(x), F_46(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_46(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_22(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_22(x))
Eq(F_64(x), F_61(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_18(x) + F_46(x))
Eq(F_68(x), F_69(x) + F_80(x))
Eq(F_69(x), F_28(x) + F_70(x) + F_77(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_31(x) + F_72(x))
Eq(F_72(x), F_22(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_63(x) + F_76(x))
Eq(F_76(x), F_73(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_25(x) + F_79(x))
Eq(F_79(x), F_69(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_14(x))
Pack JSON:
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Specification JSON:
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