OFFSET
0,4
COMMENTS
An example of a sextic divisibility sequence whose characteristic polynomial has degree 6 and a 12-element dihedral Galois group. This example has a field and polynomial discriminant of 98000, which is one of the smallest possible.
REFERENCES
Found by Noam D. Elkies and described in an email from Elkies to R. K. Guy, Jan 18 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300 (corrected by Ray Chandler, Jan 19 2019)
E. L. Roettger, H. C. Williams, and R. K. Guy, Some extensions of the Lucas functions, Number Theory and Related Fields: In Memory of Alf van der Poorten, Series: Springer Proceedings in Mathematics & Statistics, Vol. 43, J. Borwein, I. Shparlinski, W. Zudilin (Eds.) 2013.
Index entries for linear recurrences with constant coefficients, signature (-1,2,5,2,-1,-1).
FORMULA
a(n) = -a(-n) for all n in Z. - Michael Somos, Dec 30 2022
EXAMPLE
G.f. = x + x^2 + 2*x^3 + 7*x^4 + 5*x^5 + 20*x^6 + 27*x^7 + 49*x^8 + 106*x^9 + ... - Michael Somos, Dec 30 2022
MATHEMATICA
CoefficientList[ Series[(x^5 + 2x^4 + x^3 + 2x^2 + x)/(x^6 + x^5 - 2x^4 - 5x^3 - 2x^2 + x + 1), {x, 0, 42}], x] (* Robert G. Wilson v, Jun 26 2011 *)
a[1] = 0; a[2] = 1; a[3] = 1; a[4] = 2; a[5] = 7; a[6] = 5; a[n_Integer] := a[n] = -a[n - 6] - a[n - 5] + 2 a[n - 4] + 5 a[n - 3] + 2 a[n - 2] - a[n - 1] (* Or *)
LinearRecurrence[{-1, 2, 5, 2, -1, -1}, {0, 1, 1, 2, 7, 5}, 43] (* Roger L. Bagula, Mar 16 2012 *)
a[ n_] := a[n] = Sign[n]*With[{m = Abs[n]}, If[ m<4, {0, 1, 1, 2}[[m+1]], -a[m-1] +2*a[m-2] +5*a[m-3] +2*a[m-4] -a[m-5] -a[m-6]]]; (* Michael Somos, Dec 30 2022 *)
PROG
(Maxima) makelist(coeff(taylor(x*(x^4+2*x^3+x^2+2*x+1)/(x^6+x^5-2*x^4-5*x^3-2*x^2+x+1), x, 0, n), x, n), n, 1, 42); /* Bruno Berselli, Jun 05 2011 */
(PARI) Vec((x^5+2*x^4+x^3+2*x^2+x)/(x^6+x^5-2*x^4-5*x^3-2*x^2+x+1)+O(x^99)) \\ Charles R Greathouse IV, Jun 06 2011
(PARI) {a(n) = sign(n)*polcoeff((x^5 + 2*x^4 + x^3 + 2*x^2 + x)/(x^6 + x^5 - 2*x^4 - 5*x^3 - 2*x^2 + x + 1) + x*O(x^abs(n)), abs(n))}; /* Michael Somos, Dec 30 2022 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 20 2011
STATUS
approved