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 récurrente 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
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
Edited by Michael Somos, Aug 02 2002
STATUS
approved