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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).

This is a generalized Lucas sequence of order 3 as defined by Roettger, Section 3.3. - Peter Bala, Mar 04 2014

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 to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (-3,-5,-5,-5,-3,-1).

FORMULA

G.f.: x(1+2x+3x^2+2x^3+x^4)/(1+3x+5x^2+5x^3+5x^4+3x^5+x^6).

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

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 *)

PROG

(PARI) a(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^n), n)

(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

N. J. A. Sloane.

EXTENSIONS

Edited by Michael Somos, Aug 02 2002

STATUS

approved

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Last modified October 16 20:34 EDT 2018. Contains 316275 sequences. (Running on oeis4.)