A285870 a(n) = floor(n/2) - floor((n+1)/6), n >= 0.

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

Author

Wolfdieter Lang, May 12 2017

Keywords

• nonn
• easy

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)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Crossrefs

Cf. A049310, A059169, A285869, A285872.

Programs

Magma

[Floor(n/2)-Floor((n+1)/6): n in [0..100]]; // Vincenzo Librandi, May 15 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 *)

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

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