0132_0231_1230_2031_2310
Counting sequence:
1, 1, 2, 6, 19, 57, 160, 425, 1082, 2667, 6412, 15117, 35086, 80399, 182288, 409617, 913426, 2023443, 4456468, 9764885, 21299222, 46268439, 100139032, 216006681, 464519194, 996147227, 2130706460, 4546625565, 9680453662, 20568866847, 43620761632, 92341796897, 195152576546, 411779989539, 867583393828, 1825361100837, 3835405795366, 8048768712743, 16870631538728, 35321811042345, 73873437491242, 154343944749099, 322156906938412, 671801604571181, 1399678302158894, 2913705813606447, 6060508092301360, 12596005207801905, 26159580648046642, 54289486133067827, 112589990684262452, 233342755693133877, 483292535012196406, 1000362067229671479, 2069404028776742968, 4278419646001971257, 8840566068528283706, 18257592889359990843, 37686121681836310588, 77750143966924243005, 160328146734389657662, 330456126257937514559, 680800148470343139392, 1401952549601925922881, 2885762526030937980994, 5937545748725261926467, 12211744576795723169860, 25106018684318699749445, 51595543174165615870022, 105994991447535083585607, 217671580069772709068872, 446853928441540178346057, 917024541392249229934666, 1881272747613194912006219, 3858173416504500139589708, 7909963858806655732940877, 16211884135691492018618446, 33217125840505084433137743, 68039856285185848238932048, 139328700710586012384886865, 285230935565326570907238482, 583760055146414062736244819, 1194418709779253624609701972, 2443239081441165562081181781, 4996490412467262379060625494, 10215423175743616526267187287, 20880566756383875105525071960, 42670245729117951350429188185, 87178058704050139046411763802, 178069937525956418917520900187, 363644886540080455751617740892, 742454538561406819870749753437, 1515548093095126801545252831326, 3092993188154522616835461873759, 6311018320276868641435735294048, 12874576408567954858950891929697, 26259184113321486391160223039586, 53548334339328409171036517433443, 109176407944656257203903563563108, 222551908502568524300264956493925, 453581230394163332723039115673702
Generating function in Maple syntax:
(x^6+x^5-11*x^4+23*x^3-19*x^2+7*x-1)/(x-1)^2/(2*x-1)^3
Generating function in latex syntax:
\frac{x^{6}+x^{5}-11 x^{4}+23 x^{3}-19 x^{2}+7 x -1}{\left(x -1\right)^{2} \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(x**6 + x**5 - 11*x**4 + 23*x**3 - 19*x**2 + 7*x - 1)/((x - 1)**2*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
-(x-1)^2*(2*x-1)^3*F(x)+x^6+x^5-11*x^4+23*x^3-19*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{2} \left(2 x -1\right)^{3} F \! \left(x \right)+x^{6}+x^{5}-11 x^{4}+23 x^{3}-19 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/32*(n^2+15*n-50)*2^n+n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(n^{2}+15 n -50\right) 2^{n}}{32}+n +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) = 57
a(6) = 160
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)+1-n, 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) = 57
a \! \left(6\right) = 160
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+1-n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19048/
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[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[33,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[14,x]+F[27,x]
F[27,x] = F[18,x]+F[28,x]+F[30,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[14,x]+F[27,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[30,x]
F[33,x] = F[18,x]+F[34,x]+F[62,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[45,x]
F[36,x] = F[37,x]+F[6,x]
F[37,x] = F[18,x]+F[19,x]+F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[13,x]+F[40,x]
F[40,x] = F[32,x]+F[41,x]
F[41,x] = 2*F[18,x]+F[42,x]+F[44,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[17,x]+F[41,x]
F[44,x] = F[12,x]*F[40,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[18,x]+F[38,x]+F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[46,x]+F[6,x]
F[49,x] = F[18,x]+F[42,x]+F[50,x]+F[52,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[37,x]+F[49,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[40,x]+F[54,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[32,x]+F[55,x]
F[58,x] = 3*F[18,x]+F[59,x]+F[61,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[41,x]+F[58,x]
F[61,x] = F[12,x]*F[54,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[73,x]
F[64,x] = F[23,x]+F[65,x]
F[65,x] = F[18,x]+F[66,x]+F[72,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[20,x]+F[68,x]
F[68,x] = F[17,x]+F[69,x]
F[69,x] = F[18,x]+F[42,x]+F[44,x]+F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[17,x]+F[69,x]
F[72,x] = F[12,x]*F[64,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[30,x]
F[75,x] = 2*F[18,x]+F[42,x]+F[76,x]
F[76,x] = F[12,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_{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_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \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_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{62}\! \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_{45}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{6}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{17}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{46}\! \left(x \right)+F_{6}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{40}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{32}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{41}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{73}\! \left(x \right)
F_{64}\! \left(x \right) = F_{23}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{72}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{20}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{17}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{17}\! \left(x \right)+F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{30}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \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_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_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_33(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_14(x) + F_27(x))
Eq(F_27(x), F_18(x) + F_28(x) + F_30(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_14(x) + F_27(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_30(x))
Eq(F_33(x), F_18(x) + F_34(x) + F_62(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_45(x))
Eq(F_36(x), F_37(x) + F_6(x))
Eq(F_37(x), F_18(x) + F_19(x) + F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_13(x) + F_40(x))
Eq(F_40(x), F_32(x) + F_41(x))
Eq(F_41(x), 2*F_18(x) + F_42(x) + F_44(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_17(x) + F_41(x))
Eq(F_44(x), F_12(x)*F_40(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_18(x) + F_38(x) + F_47(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_46(x) + F_6(x))
Eq(F_49(x), F_18(x) + F_42(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_37(x) + F_49(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_40(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_32(x) + F_55(x))
Eq(F_58(x), 3*F_18(x) + F_59(x) + F_61(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_41(x) + F_58(x))
Eq(F_61(x), F_12(x)*F_54(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_73(x))
Eq(F_64(x), F_23(x) + F_65(x))
Eq(F_65(x), F_18(x) + F_66(x) + F_72(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_20(x) + F_68(x))
Eq(F_68(x), F_17(x) + F_69(x))
Eq(F_69(x), F_18(x) + F_42(x) + F_44(x) + F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_17(x) + F_69(x))
Eq(F_72(x), F_12(x)*F_64(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_30(x))
Eq(F_75(x), 2*F_18(x) + F_42(x) + F_76(x))
Eq(F_76(x), F_12(x)*F_73(x))
Pack JSON:
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Specification JSON:
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