0132_0213_0321_1230_2103
Counting sequence:
1, 1, 2, 6, 19, 51, 119, 273, 624, 1415, 3189, 7176, 16140, 36284, 81544, 183243, 411764, 925248, 2079033, 4671565, 10496937, 23586431, 52998254, 119086021, 267583891, 601255578, 1351009050, 3035689810, 6821133122, 15326947037, 34439337414, 77384488770, 173881367951, 390707887295, 877912653803, 1972651826981, 4432508420508, 9959756014235, 22379368622033, 50285984837828, 112991582283496, 253889780782832, 570485159011372, 1281868516522119, 2880332411272824, 6472048180056444, 14542560254863637, 32676836278510937, 73424184631829117, 164982645287305587, 370712638927929402, 832983738511335321, 1871697470663294511, 4205666040909994998, 9450045813971949238, 21234060198190599014, 47712500169443152326, 107209014723104954489, 240896469417462462346, 541289453388586726910, 1216266361471530961739, 2732925706914186188107, 6140828321911316611103, 13798315989265288448633, 31004534593527707320328, 69666556854409386478255, 156539332313081191828269, 351740686887044062814528, 790354149232759930979140, 1775909653039482732944612, 3990432768424681334053904, 8966421040656085470073347, 20147365196697369541256060, 45270718665626143218531640, 101722381487293570508246065, 228568116443515914693767781, 513587895708699256677250065, 1154021526373620857540021927, 2593062832012425057063526214, 5826559294689771714989824365, 13092159895018347629503153099, 29417816252714041916932604434, 66101233105756659748378537682, 148528122569209013784186526778, 333739661991460645399818986890, 749906213446373644835445962965, 1685023966314998921286524386126, 3786214484084910842008675748410, 8507546721038446960468429920071, 19116283959846805841659011202519, 42953900156647147801777776576787, 96516537552102654484746134436109, 216870691301005650929611034246580, 487304019997466808542190426352579, 1094962193743836613529245752515721, 2460357716184134384671070897137532, 5528373606114638574329197120438304, 12422142734669574919800219385498520, 27912301359269654029258564994297908, 62718371847094244403988152253656127, 140926902318788567917434650116111744
Generating function in Maple syntax:
(3*x^7-7*x^6+2*x^5+3*x^4+3*x^2-3*x+1)/(x^2+1)/(x^3-x^2-2*x+1)/(x-1)^2
Generating function in latex syntax:
\frac{3 x^{7}-7 x^{6}+2 x^{5}+3 x^{4}+3 x^{2}-3 x +1}{\left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(3*x**7 - 7*x**6 + 2*x**5 + 3*x**4 + 3*x**2 - 3*x + 1)/((x - 1)**2*(x**2 + 1)*(x**3 - x**2 - 2*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2+1)*(x^3-x^2-2*x+1)*(x-1)^2*F(x)-3*x^7+7*x^6-2*x^5-3*x^4-3*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)-3 x^{7}+7 x^{6}-2 x^{5}-3 x^{4}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/15288*((-183*I*3^(1/2)+23)*(-28+84*I*3^(1/2))^(2/3)-252*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)+1624*(-28+84*I*3^(1/2))^(1/3)-1960)*(1/168*(I*3^(1/2)+5)*(-28+84*I*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/15288*((103*I*3^(1/2)+263)*(-28+84*I*3^(1/2))^(2/3)-686*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-1190*(-28+84*I*3^(1/2))^(1/3)-1960)*(1/84*(I*3^(1/2)-2)*(-28+84*I*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/15288*((80*I*3^(1/2)-286)*(-28+84*I*3^(1/2))^(2/3)+938*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-434*(-28+84*I*3^(1/2))^(1/3)-1960)*(1/6*(-28+84*I*3^(1/2))^(1/3)+1/3-1/168*(-28+84*I*3^(1/2))^(2/3)-1/56*I*3^(1/2)*(-28+84*I*3^(1/2))^(2/3))^(-n)-n-8/13*cos(1/2*n*Pi)+14/13*sin(1/2*n*Pi)-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-183 \,\mathrm{I} \sqrt{3}+23\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-252 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}+1624 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-1960\right) \left(\frac{\left(\mathrm{I} \sqrt{3}+5\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{15288}\\+\\\frac{\left(\left(103 \,\mathrm{I} \sqrt{3}+263\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-686 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-1190 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-1960\right) \left(\frac{\left(\mathrm{I} \sqrt{3}-2\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{84}+\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{15288}\\+\\\frac{\left(\left(80 \,\mathrm{I} \sqrt{3}-286\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+938 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-434 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-1960\right) \left(\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{56}\right)^{-n}}{15288}\\-n -\frac{8 \cos \left(\frac{n \pi}{2}\right)}{13}+\frac{14 \sin \left(\frac{n \pi}{2}\right)}{13}-1 & \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) = 51
a(6) = 119
a(7) = 273
a(n+2) = a(n)-a(n+1)-2*a(n+4)+a(n+5)-2*n-8, 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) = 51
a \! \left(6\right) = 119
a \! \left(7\right) = 273
a \! \left(n +2\right) = a \! \left(n \right)-a \! \left(n +1\right)-2 a \! \left(n +4\right)+a \! \left(n +5\right)-2 n -8, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18262/
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[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[62,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[33,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[30,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[26,x]*F[4,x]
F[29,x] = F[14,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[16,x]+F[32,x]
F[32,x] = F[27,x]
F[33,x] = F[34,x]+F[41,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[2,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[37,x]*F[4,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[19,x]+F[46,x]+F[53,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[50,x]
F[48,x] = F[11,x]+F[49,x]
F[49,x] = F[15,x]+F[19,x]+F[24,x]
F[50,x] = F[39,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[44,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[2,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[11,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[26,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[54,x]+F[64,x]
F[64,x] = F[45,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,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_{11}\! \left(x \right)+F_{14}\! \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_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{62}\! \left(x \right)
F_{19}\! \left(x \right) = 0
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_{33}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \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_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right)+F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{50}\! \left(x \right) = F_{39}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{2}\! \left(x \right)+F_{55}\! \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_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{11}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{26}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{54}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{45}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \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_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_62(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_33(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_30(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_26(x)*F_4(x))
Eq(F_29(x), F_14(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_16(x) + F_32(x))
Eq(F_32(x), F_27(x))
Eq(F_33(x), F_34(x) + F_41(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_2(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_37(x)*F_4(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_19(x) + F_46(x) + F_53(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_50(x))
Eq(F_48(x), F_11(x) + F_49(x))
Eq(F_49(x), F_15(x) + F_19(x) + F_24(x))
Eq(F_50(x), F_39(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_44(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_2(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_11(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_26(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_54(x) + F_64(x))
Eq(F_64(x), F_45(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_14(x))
Pack JSON:
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Specification JSON:
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