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 A005120 A sixth-order linear divisibility sequence: a(n+6) = -3*a(n+5) - 5*a(n+4) - 5*a(n+3) - 5*a(n+2) - 3*a(n+1) - a(n). (Formerly M3770) 1
 0, 1, -1, 1, -1, -1, 5, -8, 7, 1, -19, 43, -55, 27, 64, -211, 343, -307, -85, 911, -1919, 2344, -989, -3151, 9625, -15049, 12609, 5671, -42496, 85609, -100225, 33977, 154007, -437009, 657901, -513512, -335665, 1974097, -3808891, 4265379 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS This is a divisibility sequence. If d divides n then a(d) divides a(n). - Michael Somos, Aug 02 2002 This is a generalized Lucas sequence of order 3 as defined by Roettger, Section 3.3. - Peter Bala, Mar 04 2014 The sequence is denoted by C(n) and its expression in terms of the roots of the cubic x^3 + x^2 - 1 = 0 is given in Williams 1998 page 454. Table 17.4.1 Values of C(n) for n=-2 to n=30 is on page 455 and he notes that a(20) == a(25) == 0 (mod 101). - Michael Somos, Nov 13 2018 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). H. C. Williams, Edouard Lucas and Primality Testing, Wiley, 1998, p. 455. Math. Rev. 2000b:11139 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 M. Duboue, Une suite recurrente remarquable, Europ. J. Combin., 4 (1983), 205-214. E. L. F. Roettger, A cubic extension of the Lucas functions, Thesis, Dept. of Mathematics and Statistics, Univ. of Calgary, 2009 Index entries for linear recurrences with constant coefficients, signature (-3,-5,-5,-5,-3,-1). FORMULA G.f.: x * (1 + 2*x + 3*x^2 + 2*x^3 + x^4) / (1 + 3*x + 5*x^2 + 5*x^3 + 5*x^4 + 3*x^5 + x^6). - Michael Somos, Aug 02 2002 a(n) = (a^n - b^n)*(b^n - c^n)*(c^n - a^n)/((a - b)*(b - c)*(c - a)), where a, b, c denote the roots of the cubic equation x^3 + x^2 - 1 = 0. - Peter Bala, Mar 04 2014 a(n) = -3*a(n-1) - 5*a(n-2) - 5*a(n-3) - 5*a(n-4) - 3*a(n-5) - a(n-6) for n>5. - Vincenzo Librandi, Jun 20 2014 a(n) = -a(-n) for all n in Z. - Michael Somos, Nov 13 2018 EXAMPLE G.f. = x - x^2 + x^3 - x^4 - x^5 + 5*x^6 - 8*x^7 + 7*x^8 + ... - Michael Somos, Nov 13 2018 MATHEMATICA LinearRecurrence[{-3, -5, -5, -5, -3, -1}, {0, 1, -1, 1, -1, -1}, 40] (* Harvey P. Dale, Jun 19 2014 *) CoefficientList[Series[x (1 + 2 x + 3 x^2 + 2 x^3 + x^4)/(1 + 3 x + 5 x^2 + 5 x^3 + 5 x^4 + 3 x^5 + x^6), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 20 2014 *) a[ n_] := Sign[n] SeriesCoefficient[ x (1 + x + x^2)^2 / (1 + 3 x + 5 x^2 + 5 x^3 + 5 x^4 + 3 x^5 + x^6), {x, 0, Abs @ n}]; (* Michael Somos, Nov 13 2018 *) a[ n_] := Module[ {a, b, c}, {a, b, c} = Table[ Root[#^3 + #^2 - 1 &, k], {k, 3}]; (a^n - b^n) (b^n - c^n) (c^n - a^n) / ((a - b) (b - c) (c - a)) // FullSimplify]; (* Michael Somos, Nov 13 2018 *) PROG (PARI) {a(n) = sign(n) * polcoeff( x*(1 + 2*(x + x^3) + 3*x^2 + x^4) / (1 + 3*(x + x^5) + 5*(x^2  + x^3 + x^4) + x^6) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, Aug 02 2002 */ (MAGMA) I:=[0, 1, -1, 1, -1, -1]; [n le 6 select I[n] else -3*Self(n-1)-5*Self(n-2)-5*Self(n-3)-5*Self(n-4)-3*Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Jun 20 2014 CROSSREFS Cf. A001608. Sequence in context: A145432 A070371 A199444 * A133731 A021067 A047914 Adjacent sequences:  A005117 A005118 A005119 * A005121 A005122 A005123 KEYWORD sign,easy AUTHOR EXTENSIONS Edited by Michael Somos, Aug 02 2002 STATUS approved

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Last modified January 18 00:34 EST 2019. Contains 319255 sequences. (Running on oeis4.)