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A285870
a(n) = floor(n/2) - floor((n+1)/6), n >= 0.
3
0, 0, 1, 1, 2, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 5, 6, 5, 6, 6, 7, 7, 8, 7, 8, 8, 9, 9, 10, 9, 10, 10, 11, 11, 12, 11, 12, 12, 13, 13, 14, 13, 14, 14, 15, 15, 16, 15, 16, 16, 17, 17, 18, 17, 18, 18, 19, 19, 20, 19, 20, 20, 21, 21, 22, 21, 22, 22, 23, 23, 24
OFFSET
0,5
COMMENTS
This is the number of integers k in the (left-sided open) interval ((n+1)/6, floor(n/2)]. This sequence is used in A285872(n), the number of zeros of Chebyshev's S(n, x) polynomial (A049310) in the open interval (-sqrt(3), +sqrt(3)).
FORMULA
a(n) = floor(n/2) - floor((n+1)/6), n >= 0.
G.f.: (x^2/((1+x)*(1-x)^2))*(1-x^3/((1+x+x^2)*(1-x+x^2))).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>6. - Colin Barker, May 18 2017
MATHEMATICA
Table[Floor[n/2] - Floor[(n + 1)/6], {n, 0, 60}] (* or *)
CoefficientList[Series[(x^2/((1 + x) (1 - x)^2)) (1 - x^3/((1 + x + x^2) (1 - x + x^2))), {x, 0, 60}], x] (* Michael De Vlieger, May 13 2017 *)
PROG
(Magma) [Floor(n/2)-Floor((n+1)/6): n in [0..100]]; // Vincenzo Librandi, May 15 2017
(PARI) concat(vector(2), Vec(x^2*(1 + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^100))) \\ Colin Barker, May 18 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, May 12 2017
STATUS
approved