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.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

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). - Richard Choulet, Aug 14 2007

From Peter Bala, Sep 15 2019: (Start)

This is a linear divisibility sequence of order 6. It is a particular case of a family of divisibility sequences studied by Roettger et al. The o.g.f. has the form x*d/dx(f(x)/(x^3*f(1/x))) where f(x) = x^3 - x - 1.

More generally, if f(x) = 1 + P*x + Q*x^2 + x^3 or f(x) = -1 + P*x + Q*x^2 + x^3, where P and Q are integers, then the rational function x*d/dx(f(x)/(x^3*f(1/x))) is the generating function for a linear divisibility sequence of order 6. Cf. A001351. There are corresponding results when f(x) is a monic quartic polynomial with constant term 1. (End)

Resultant of the (s_3, s_3+n) pair where s_n(X) is X^n-X-1. See Rush link. - Michel Marcus, Sep 30 2019

Examples

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

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
• Peter Bala, Some linear divisibility sequences of order 6
• 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; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 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.
• David E. Rush, Degree n Relatives of the Golden Ratio and Resultants of the Corresponding Polynomials, Fib. Q. 50(4), 2012, 313-325. See p. 319.
• Index entries for linear recurrences with constant coefficients, signature (-1,1,3,1,-1,-1).

Crossrefs

Cf. A001608, A078712, A104499, A001351.

Programs

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

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 *)

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

PARI

L3(n) = polsym(x^3-x-1, n)[n+1]; \\ A001608
a(n) = my(L3n=L3(n)); L3n - matdet([L3n, L3(2*n);1, L3n])/2; \\ Michel Marcus, Sep 30 2019

Formula

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

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

a(n) = (alpha^n - 1)*(beta^n - 1)*(gamma^n - 1) where alpha, beta and gamma are the zeros of x^3 - x - 1. - Peter Bala, Sep 15 2019

Extensions

More terms from James Sellers, Dec 23 1999