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)

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

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 to divisibility sequences

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: A365793 A070371 A199444 * A362975 A133731 A021067

Adjacent sequences: A005117 A005118 A005119 * A005121 A005122 A005123

Keyword

sign,easy

Author

N. J. A. Sloane

Extensions

Edited by Michael Somos, Aug 02 2002

Status

approved