%I #40 Dec 12 2023 07:45:38
%S 1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,
%T 0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,
%U 1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0
%N Period 5: repeat [1,0,0,1,0].
%C Used by R. J. Baxter in studying the Rogers-Ramanujan identities.
%D Andrews, George E., q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra. CBMS Regional Conference Series in Mathematics, 66. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. xii+130 pp. ISBN: 0-8218-0716-1 MR0858826 (88b:11063). See p. 105.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,1).
%F a(n) = A198517(n+2).
%F a(n) = 2 + floor(n/5) - ceiling(n/5) + floor((n - 3)/5) - ceiling((n - 3)/5). - _Wesley Ivan Hurt_, Mar 13 2014
%F G.f.: -(x + 1)*(x^2 - x + 1)/((x - 1)*(x^4 + x^3 + x^2 + x + 1)). - _Colin Barker_, Mar 14 2014
%F a(n) = floor(((2*n - 2) mod 5)/3). - _Wesley Ivan Hurt_, Apr 30 2015
%F a(n) = (2/5)*(1 + cos(2*(n-3)*Pi/5) + cos(4*(n-3)*Pi/5) + cos(2*n*Pi/5) + cos(4*n*Pi/5)). - _Wesley Ivan Hurt_, Sep 26 2018
%p A232990:=n->2 + floor(n/5) - ceil(n/5) + floor((n-3)/5) - ceil((n-3)/5); # _Wesley Ivan Hurt_, Mar 13 2014
%t Table[2 + Floor[n/5] - Ceiling[n/5] + Floor[(n - 3)/5] - Ceiling[(n - 3)/5], {n, 0, 100}] (* _Wesley Ivan Hurt_, Mar 13 2014 *)
%o (PARI) Vec(-(x+1)*(x^2-x+1)/((x-1)*(x^4+x^3+x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Mar 14 2014
%o (PARI) a(n)=((2*n-2)%5)\3 \\ _Charles R Greathouse IV_, Apr 30 2015
%o (Magma) /* By definition: */ &cat [[1,0,0,1,0]: n in [0..20]]; // _Bruno Berselli_, Feb 18 2015
%o (Magma) [(((n+1) mod 5) mod 3) mod 2: n in [0..100]]; // _Vincenzo Librandi_, Feb 18 2015
%Y Cf. A198517, A232991.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_, Dec 13 2013