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A001945 a(n+6) = -a(n+5) + a(n+4) + 3a(n+3) + a(n+2) - a(n+1) - a(n). a(n) = sign(n) if abs(n)<=3.
(Formerly M3730 N1525)
6
0, 1, 1, 1, 5, 1, 7, 8, 5, 19, 11, 23, 35, 27, 64, 61, 85, 137, 133, 229, 275, 344, 529, 599, 875, 1151, 1431, 2071, 2560, 3481, 4697, 5953, 8245, 10649, 14111, 19048, 24605, 33227, 43739, 57591, 77275, 101107, 134848, 178709, 235405, 314089, 413909 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

It seems likely that this sequence contains infinitely many primes. In the paper by Einsiedler, Everest, Ward the heuristics for the Mersenne sequence are adapted to argue that approximately c*log(N) of the first N terms should be prime, where c is constant. Numerical evidence is provided to support this. - Graham Everest (g.everest(AT)uea.ac.uk), Mar 01 2001

Comments from Richard Choulet, Aug 14 2007: For n>=4 a(n-4) is the resultant of the polynomials x^3-x-1 and x^(n+1)-x^n-1. For n=4 in fact the result is 0 as we see from the identity x^5-x^4-1=(x^3-x-1)(x^2-x+1). The characteristic polynomial of the sequence is x^6+x^5-x^4-3x^3-x^2+x+1 = (x^3-x-1)*(x^3+x^2-1).

a(A104499(n+1)) = A204138(n). - Reinhard Zumkeller, Jan 11 2012

REFERENCES

G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.

M. Hall, A slowly increasing arithmetic sequence, J. London Math. Soc., 8 (1933), 162-166.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Manfred Einsiedler, Graham Everest and Thomas Ward, Primes in sequences associated to polynomials (after Lehmer), LMS J. Comput. Math. 3 (2000), 125-139.

G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

E. L. Roettger, H. C. Williams, 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, 1, 3, 1, -1, -1).

FORMULA

G.f.: A(x) = (x^5+2x^4+x^3+2x^2+x)/(x^6+x^5-x^4-3x^3-x^2+x+1) - Ralf Stephan, Dec 15 2002

a(n) ~ r1^n-2*real(r2^n), with r1=1.324717957 the inverse real root of x^3+x^2-1=0 and r2=(0.87744+0.7448617i) one inverse complex root of x^3-x-1=0. With n>9, a(n) = round(r1^n-2*real(r2^n)). - Ralf Stephan, Dec 17 2002

a(n) = A001608(n) + A078712(n). - Ralf Stephan, Dec 27 2002

a(-n) = -a(n). - Michael Somos, Apr 25 2014

EXAMPLE

G.f. = x + x^2 + x^3 + 5*x^4 + x^5 + 7*x^6 + 8*x^7 + 5*x^8 + 19*x^9 + ...

MAPLE

A001945:=z*(1+2*z+z**2+2*z**3+z**4)/(z**3-z-1)/(z**3+z**2-1); # conjectured by Simon Plouffe in his 1992 dissertation

MATHEMATICA

a[0] = 0; a[1] = a[2] = a[3] = a[5] = 1; a[4] = 5; a[n_] := a[n] = -a[n - 1] + a[n - 2] + 3a[n - 3] + a[n - 4] - a[n - 5] - a[n - 6]; Table[ a[n], {n, 0, 46}] (* Robert G. Wilson v, Mar 10 2005 *)

LinearRecurrence[{-1, 1, 3, 1, -1, -1}, {0, 1, 1, 1, 5, 1}, 50] (* T. D. Noe, Jan 11 2012 *)

a[ n_] := Sign[n] SeriesCoefficient[ x * (1 + 2 x + x^2 + 2 x^3 + x^4) / (1 + x - x^2 - 3 x^3 - x^4 + x^5 + x^6), {x, 0, Abs @ n}]; (* Michael Somos, Apr 25 2014 *)

PROG

(Haskell)

import Data.List (zipWith6)

a001945 n = a001945_list !! n

a001945_list = 0 : 1 : 1 : 1 : 5 : 1 : zipWith6

   (\u v w x y z -> - u + v + 3*w + x - y - z)

     (drop 5 a001945_list) (drop 4 a001945_list) (drop 3 a001945_list)

     (drop 2 a001945_list) (drop 1 a001945_list) (drop 0 a001945_list)

-- Reinhard Zumkeller, Jan 11 2012

(PARI) {a(n) = sign(n) * polcoeff( x * (1 + 2*x + x^2 + 2*x^3 + x^4) / (1 + x - x^2 - 3*x^3 - x^4 + x^5 + x^6) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, Apr 25 2014 */

(PARI) a(n)=([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; -1, -1, 1, 3, 1, -1]^n*[0; 1; 1; 1; 5; 1])[1, 1] \\ Charles R Greathouse IV, Jul 19 2016

CROSSREFS

Cf. A001608, A078712, A104499.

Sequence in context: A200638 A322104 A100122 * A233091 A286941 A051854

Adjacent sequences:  A001942 A001943 A001944 * A001946 A001947 A001948

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Dec 23 1999

STATUS

approved

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Last modified September 15 12:44 EDT 2019. Contains 327078 sequences. (Running on oeis4.)