0123_0231_1320_2013_2130
Counting sequence:
1, 1, 2, 6, 19, 51, 123, 276, 586, 1195, 2366, 4584, 8743, 16489, 30851, 57402, 106394, 196685, 362962, 669026, 1232227, 2268407, 4174563, 7680888, 14130418, 25993383, 47813246, 87946740, 161764295, 297536541, 547261275, 1006577358, 1851392082, 3405249401, 6263239426, 11519903518, 21188417107, 38971587099, 71679937195, 131839973436, 242491532474, 446011480707, 820343027230, 1508846084192, 2775200639239, 5104389801265, 9388436578963, 17268027077570, 31760853519914, 58417317242757, 107446197910930, 197624368748858, 363487883982563, 668558450727327, 1229670703548883, 2261717038354272, 4159946192731554, 7651333934741567, 14072997165940254, 25884277293532460, 47608608394339815, 87565882853944741, 161058768541956139, 296233259790386966, 544857911186441506, 1002149939518945905, 1843241110495943554, 3390248961201508278, 6235640011216583443, 11469130082914229635, 21095019055332524635, 38799789149463550180, 71363938287710526378, 131258746492506832859, 241422473929681151102, 444045158709898762328, 816726379132087008871, 1502194011771667195769, 2762965549613653251619, 5081885940517407752394, 9347045501902728507706, 17191896992033789831741, 31620828434453926424274, 58159770928390445108882, 106972496354878161723107, 196753095717722533627847, 361885363000991140845123, 665610955073591836595400, 1224249413792305511482066, 2251745731866888489350999, 4141606100732785837871934, 7617601246391979839163876, 14010953078991654166861447, 25770160426116419844388013, 47398714751500053850920571, 87179828256608127862694110, 160348703434224601558543986, 294927246442332783272717545, 542455778133165512694532482, 997731728009722897526389198, 1835114752585221193494253139
Generating function in Maple syntax:
-(x^7+3*x^6-4*x^5+3*x^4-2*x^3+6*x^2-4*x+1)/(x^3+x^2+x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{x^{7}+3 x^{6}-4 x^{5}+3 x^{4}-2 x^{3}+6 x^{2}-4 x +1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-x**7 - 3*x**6 + 4*x**5 - 3*x**4 + 2*x**3 - 6*x**2 + 4*x - 1)/((x - 1)**4*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)*(x-1)^4*F(x)+x^7+3*x^6-4*x^5+3*x^4-2*x^3+6*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{7}+3 x^{6}-4 x^{5}+3 x^{4}-2 x^{3}+6 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/1056*(((-196*11^(1/2)+792*I)*3^(1/2)-588*I*11^(1/2)+792)*(17+3*11^(1/2)*3^(1/2))^(1/3)+2112+((239*11^(1/2)+1485*I)*3^(1/2)-717*I*11^(1/2)-1485)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/1056*(((239*11^(1/2)-1485*I)*3^(1/2)+717*I*11^(1/2)-1485)*(17+3*11^(1/2)*3^(1/2))^(2/3)+2112+((-196*11^(1/2)-792*I)*3^(1/2)+588*I*11^(1/2)+792)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/1056*((-478*11^(1/2)*3^(1/2)+2970)*(17+3*11^(1/2)*3^(1/2))^(2/3)+392*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-1584*(17+3*11^(1/2)*3^(1/2))^(1/3)+2112)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)-1/3*n^3-1/4*n^2-71/12*n-4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(-196 \sqrt{11}+792 \,\mathrm{I}\right) \sqrt{3}-588 \,\mathrm{I} \sqrt{11}+792\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+2112+\left(\left(239 \sqrt{11}+1485 \,\mathrm{I}\right) \sqrt{3}-717 \,\mathrm{I} \sqrt{11}-1485\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{1056}\\+\\\frac{\left(\left(\left(239 \sqrt{11}-1485 \,\mathrm{I}\right) \sqrt{3}+717 \,\mathrm{I} \sqrt{11}-1485\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+2112+\left(\left(-196 \sqrt{11}-792 \,\mathrm{I}\right) \sqrt{3}+588 \,\mathrm{I} \sqrt{11}+792\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{1056}\\+\\\frac{\left(\left(-478 \sqrt{11}\, \sqrt{3}+2970\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+392 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-1584 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+2112\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{1056}\\-\frac{n^{3}}{3}-\frac{n^{2}}{4}-\frac{71 n}{12}-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) = 19
a(5) = 51
a(6) = 123
a(7) = 276
a(n+3) = 2/3*n^3+1/2*n^2+a(n)+a(n+1)+a(n+2)+47/6*n+1, 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) = 123
a \! \left(7\right) = 276
a \! \left(n +3\right) = \frac{2 n^{3}}{3}+\frac{n^{2}}{2}+a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+\frac{47 n}{6}+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17507/
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[19,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[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[44,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[29,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[7,x]
F[29,x] = F[30,x]+F[35,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[7,x]
F[34,x] = F[30,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[25,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[25,x]
F[43,x] = F[40,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[70,x]
F[46,x] = F[2,x]+F[47,x]
F[47,x] = F[21,x]+F[48,x]+F[68,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[11,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[28,x]*F[4,x]
F[53,x] = F[54,x]+F[59,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[11,x]
F[58,x] = F[54,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[51,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[51,x]
F[67,x] = F[64,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[2,x]+F[54,x]
F[70,x] = F[30,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_{19}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{44}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \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_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{25}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \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_{70}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{21}\! \left(x \right)+F_{48}\! \left(x \right)+F_{68}\! \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_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{11}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{59}\! \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_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{11}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \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_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{51}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{51}\! \left(x \right)
F_{67}\! \left(x \right) = F_{64}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{2}\! \left(x \right)+F_{54}\! \left(x \right)
F_{70}\! \left(x \right) = F_{30}\! \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_19(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_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_44(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_29(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_7(x))
Eq(F_29(x), F_30(x) + F_35(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_7(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_25(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_25(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_70(x))
Eq(F_46(x), F_2(x) + F_47(x))
Eq(F_47(x), F_21(x) + F_48(x) + F_68(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_11(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_28(x)*F_4(x))
Eq(F_53(x), F_54(x) + F_59(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_11(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_51(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_51(x))
Eq(F_67(x), F_64(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_2(x) + F_54(x))
Eq(F_70(x), F_30(x))
Pack JSON:
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Specification JSON:
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