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

1,1

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);

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

Links

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.

Formula

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

Mathematica

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

Crossrefs

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

Sequence in context: A284882 A190364 A088366 * A376960 A186228 A184747

Adjacent sequences: A257981 A257982 A257983 * A257985 A257986 A257987

Keyword

nonn,easy

Author

Clark Kimberling, Jun 15 2015

Status

approved