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A257984 Nonhomogeneous Beatty sequence: ceiling((n - 1/2)*Pi). 3
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 (list; graph; refs; listen; history; text; internal format)
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
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 * A186228 A184747 A361684
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 15 2015
STATUS
approved

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Last modified April 20 06:53 EDT 2024. Contains 371799 sequences. (Running on oeis4.)