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A001945
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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)
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10
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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
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OFFSET
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0,5
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COMMENTS
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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
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
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REFERENCES
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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).
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LINKS
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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.
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FORMULA
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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) = (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
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EXAMPLE
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G.f. = x + x^2 + x^3 + 5*x^4 + x^5 + 7*x^6 + 8*x^7 + 5*x^8 + 19*x^9 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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)
(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
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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