0132_0321_1203_2301_3120
Counting sequence:
1, 1, 2, 6, 19, 47, 98, 188, 349, 645, 1204, 2282, 4391, 8555, 16822, 33288, 66145, 131777, 262952, 525206, 1049611, 2098311, 4195594, 8390036, 16778789, 33556157, 67110748, 134219778, 268437679, 536873315, 1073744414, 2147486432, 4294970281, 8589937785, 17179872592, 34359741998, 68719480595, 137438957567, 274877911282, 549755818476, 1099511632621, 2199023260661, 4398046516484, 8796093027866, 17592186050359, 35184372095067, 70368744184198, 140737488362168, 281474976717809, 562949953428785, 1125899906850424, 2251799813693382, 4503599627378971, 9007199254749815, 18014398509491162, 36028797018973508, 72057594037937845, 144115188075866157, 288230376151722412, 576460752303434546, 1152921504606858431, 2305843009213705811, 4611686018427400174, 9223372036854788496, 18446744073709564729, 36893488147419116777, 73786976294838220448, 147573952589676427358, 295147905179352840739, 590295810358705667055, 1180591620717411319234, 2361183241434822623132, 4722366482869645230461, 9444732965739290444645, 18889465931478580872532, 37778931862957161727818, 75557863725914323437895, 151115727451828646857547, 302231454903657293696342, 604462909807314587373416, 1208925819614629174727041, 2417851639229258349433761, 4835703278458516698846664, 9671406556917033397671926, 19342813113834066795321899, 38685626227668133590621287, 77371252455336267181219498, 154742504910672534362415348, 309485009821345068724806469, 618970019642690137449588125, 1237940039285380274899150844, 2475880078570760549798275682, 4951760157141521099596524751, 9903520314283042199193022275, 19807040628566084398386016702, 39614081257132168796772004928, 79228162514264337593543980745, 158456325028528675187087931737, 316912650057057350374175833072, 633825300114114700748351635086, 1267650600228229401496703238451
Generating function in Maple syntax:
-(4*x^6+6*x^5-2*x^4+2*x^3-6*x^2+4*x-1)/(2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{4 x^{6}+6 x^{5}-2 x^{4}+2 x^{3}-6 x^{2}+4 x -1}{\left(2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-4*x**6 - 6*x**5 + 2*x**4 - 2*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^3*F(x)+4*x^6+6*x^5-2*x^4+2*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+4 x^{6}+6 x^{5}-2 x^{4}+2 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,25+2^n-39/2*n+7/2*n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 25+2^{n}-\frac{39 n}{2}+\frac{7 n^{2}}{2} & \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) = 47
a(6) = 98
a(n+1) = -7/2*n^2+2*a(n)+53/2*n-41, n >= 7
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) = 98
a \! \left(n +1\right) = -\frac{7 n^{2}}{2}+2 a \! \left(n \right)+\frac{53 n}{2}-41, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19458/
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[12,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[12,x]+F[20,x]
F[20,x] = F[15,x]+F[18,x]+F[21,x]
F[21,x] = F[10,x]*F[12,x]
F[22,x] = F[23,x]+F[38,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[32,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[9,x]
F[30,x] = F[10,x]+F[31,x]
F[31,x] = F[16,x]
F[32,x] = F[23,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[27,x]
F[37,x] = F[33,x]
F[38,x] = F[15,x]+F[39,x]+F[54,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[48,x]
F[41,x] = F[42,x]+F[6,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[45,x]
F[48,x] = F[38,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[42,x]
F[53,x] = F[49,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[64,x]
F[56,x] = F[57,x]+F[60,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[1,x]+F[57,x]
F[60,x] = F[15,x]+F[61,x]+F[63,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[10,x]+F[60,x]
F[63,x] = F[12,x]*F[56,x]
F[64,x] = F[65,x]+F[71,x]
F[65,x] = F[15,x]+F[66,x]+F[69,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[10,x]+F[68,x]
F[68,x] = F[66,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[57,x]+F[68,x]
F[71,x] = F[15,x]+F[72,x]+F[76,x]+F[78,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[47,x]
F[75,x] = F[72,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[60,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[80,x]+F[82,x]
F[80,x] = F[81,x]
F[81,x] = F[12,x]*F[57,x]
F[82,x] = 2*F[15,x]+F[78,x]+F[83,x]
F[83,x] = F[12,x]*F[60,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \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_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{38}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{32}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{27}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{54}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{6}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{38}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{42}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{64}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{1}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{10}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{71}\! \left(x \right)
F_{65}\! \left(x \right) = F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{10}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{57}\! \left(x \right)+F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{15}\! \left(x \right)+F_{72}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{47}\! \left(x \right)
F_{75}\! \left(x \right) = F_{72}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{60}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{82}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{82}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{78}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_22(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_12(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_18(x) + F_21(x))
Eq(F_21(x), F_10(x)*F_12(x))
Eq(F_22(x), F_23(x) + F_38(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_32(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_9(x))
Eq(F_30(x), F_10(x) + F_31(x))
Eq(F_31(x), F_16(x))
Eq(F_32(x), F_23(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_27(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_15(x) + F_39(x) + F_54(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_48(x))
Eq(F_41(x), F_42(x) + F_6(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_45(x))
Eq(F_48(x), F_38(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_42(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_64(x))
Eq(F_56(x), F_57(x) + F_60(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_1(x) + F_57(x))
Eq(F_60(x), F_15(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_10(x) + F_60(x))
Eq(F_63(x), F_12(x)*F_56(x))
Eq(F_64(x), F_65(x) + F_71(x))
Eq(F_65(x), F_15(x) + F_66(x) + F_69(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_10(x) + F_68(x))
Eq(F_68(x), F_66(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_57(x) + F_68(x))
Eq(F_71(x), F_15(x) + F_72(x) + F_76(x) + F_78(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_47(x))
Eq(F_75(x), F_72(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_60(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_82(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_12(x)*F_57(x))
Eq(F_82(x), 2*F_15(x) + F_78(x) + F_83(x))
Eq(F_83(x), F_12(x)*F_60(x))
Pack JSON:
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Specification JSON:
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