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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).
(Formerly M3730 N1525)
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6
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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
Comments from Richard Choulet, Aug 14 2007: For n>=4 a(n) 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]
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REFERENCES
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M. Einsiedler, G. Everest, T. Ward, Primes in sequences associated to polynomials, LMS J. Comp. Math. 3 (2000), 15-29.
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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T. D. Noe, Table of n, a(n) for n=0..1000
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 S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
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FORMULA
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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
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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 *)
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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)
-- Reinhard Zumkeller, Jan 11 2012
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CROSSREFS
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Cf. A001608, A078712, A104499.
Sequence in context: A088515 A200638 A100122 * A051854 A006569 A224139
Adjacent sequences: A001942 A001943 A001944 * A001946 A001947 A001948
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James A. Sellers, Dec 23 1999
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STATUS
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approved
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