OFFSET

0,2

COMMENTS

Let r = Pi, s = r/(r-1), and t = 1/2. Let R be the ordered set {floor[(n + t)*r] : n is an integer} and let S be the ordered set {floor[(n - t)*s : n is an integer}; thus,

R = (..., -10, -9, -7, -6, -4, -3, -1, 0, 2, 3, 5, 6, 8, ...).

S = (..., -15, -11, -8, -5, -2, 1, 4, 7, 10, 14, 17, 20, ...)

By Fraenkel's theorem (Theorem XI in the cited paper); R and S partition the integers.

R is the set of integers n such that (cos n)*(cos(n + 1)) < 0;

S is the set of integers n such that (cos n)*(cos(n + 1)) > 0.

A246046 = (2,3,6,6,8,...), positive terms of R;

A062389 = (1,4,7,10,14,17,...), positive terms of S;

A258048 = (1,3,4,6,7,9,10,...), - (nonpositive terms of R).

A257984 = (2,5,8,11,15,...), - (negative terms of S);

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..10000

A. S. Fraenkel, The bracket function and complementary sets of integers, Canadian J. of Math. 21 (1969) 6-27.

FORMULA

a(n) = ceiling((n + 1/2)*Pi/(Pi - 1)).

MATHEMATICA

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 15 2015

STATUS

approved