A257984 - OEIS

%I #10 Dec 13 2023 08:50:52

%S 2,5,8,11,15,18,21,24,27,30,33,37,40,43,46,49,52,55,59,62,65,68,71,74,

%T 77,81,84,87,90,93,96,99,103,106,109,112,115,118,121,125,128,131,134,

%U 137,140,143,147,150,153,156,159,162,165,169,172,175,178,181,184

%N Nonhomogeneous Beatty sequence: ceiling((n - 1/2)*Pi).

%C 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,

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

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

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

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

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

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

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

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

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

%C A062389 and A246046 partition the positive integers, and A258048 and A257984 partition the nonnegative integers.

%H Clark Kimberling, <a href="/A257984/b257984.txt">Table of n, a(n) for n = 1..10000</a>

%H A. S. Fraenkel, <a href="http://dx.doi.org/10.4153/CJM-1969-002-7">The bracket function and complementary sets of integers</a>, Canadian J. of Math. 21 (1969) 6-27.

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

%t Table[Ceiling[(n - 1/2) Pi], {n, 1, 120}] (* A257984 *)

%t Table[Ceiling[(n + 1/2) Pi/(Pi - 1)], {n, 0, 120}]  (* A258048 *)

%Y Cf. A258048 (complement), A246046, A062380, A258833.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jun 15 2015