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).

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

Author

N. J. A. Sloane

Keywords

• sign
• easy

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

Examples

			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
		

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).

Crossrefs

Cf. A001608.

Programs

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

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 *)

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 */

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

Extensions

Edited by Michael Somos, Aug 02 2002