0,5

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 (richardchoulet(AT)yahoo.fr), 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).

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.

S. 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.

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

S. 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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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

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

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 (ralf(AT)ark.in-berlin.de), 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 (ralf(AT)ark.in-berlin.de), Dec 17 2002

a(n) = A001608(n) + A078712(n). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Dec 27 2002

A001945:=z*(1+2*z+z**2+2*z**3+z**4)/(z**3-z-1)/(z**3+z**2-1); [Conjectured by S. Plouffe in his 1992 dissertation.]

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}] (from Robert G. Wilson v Mar 10 2005)

Cf. A001608, A078712, A104499.

Sequence in context: A101263 A088515 A100122 this_sequence A051854 A006569 A126155

Adjacent sequences: A001942 A001943 A001944 this_sequence A001946 A001947 A001948

nonn,nice,easy

njas

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 23 1999