A258048 Nonhomogeneous Beatty sequence: ceiling((n + 1/2)*Pi/(Pi- 1))

1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97

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);

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 = 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

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

Crossrefs

Cf. A257984 (complement), A246046, A062380, A258833.

Sequence in context: A186495 A184746 A186227 * A185543 A026322 A049624

Adjacent sequences: A258045 A258046 A258047 * A258049 A258050 A258051

Keyword

nonn,easy

Author

Clark Kimberling, Jun 15 2015

Status

approved