A258048 -id:A258048 - cp's OEIS frontend

A257984 Nonhomogeneous Beatty sequence: ceiling((n - 1/2)*Pi).

2, 5, 8, 11, 15, 18, 21, 24, 27, 30, 33, 37, 40, 43, 46, 49, 52, 55, 59, 62, 65, 68, 71, 74, 77, 81, 84, 87, 90, 93, 96, 99, 103, 106, 109, 112, 115, 118, 121, 125, 128, 131, 134, 137, 140, 143, 147, 150, 153, 156, 159, 162, 165, 169, 172, 175, 178, 181, 184

Offset: 1

Clark Kimberling, Jun 15 2015

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 = (2,3,5,6,8,...), positive terms of R;
A062389 = (1,4,7,10,14,17,...), positive terms of S;
A258048 = (1,3,4,6,7,9,10,...), - (negative terms of R);
A257984 = (2,5,8,11,15,...), - (negative terms of S).
A062389 and A246046 partition the positive integers, and A258048 and A257984 partition the positive integers.

Clark Kimberling, Table of n, a(n) for n = 1..10000
A. S. Fraenkel, The bracket function and complementary sets of integers, Canadian J. of Math. 21 (1969) 6-27.

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

Table[Ceiling[(n - 1/2) Pi], {n, 1, 120}] (* A257984 *)
Table[Ceiling[(n + 1/2) Pi/(Pi - 1)], {n, 0, 120}]  (* A258048 *)

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

cp's OEIS Frontend

A257984 Nonhomogeneous Beatty sequence: ceiling((n - 1/2)*Pi).

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