A258833 Nonhomogeneous Beatty sequence: ceiling((n + 1/4)*sqrt(2)).

1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 31, 32, 33, 35, 36, 38, 39, 40, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 69, 70, 72, 73, 74, 76, 77, 79, 80, 81, 83, 84, 86, 87, 89, 90, 91, 93

Offset: 0

Author

Clark Kimberling, Jun 12 2015

Keywords

• nonn
• easy

Comments

Complement of A258834.

Let r = sqrt(2) and s = r/(r-1) = 2 + sqrt(2). Let R be the ordered set {floor[(n + 1/4)*r] : n is an integer} and let S be the ordered set {floor[(n - 1/4)*s : n is an integer}; thus,

R = (..., -8, -7, -5, -4, -2, -1, 1, 2, 3, 5, 6, ...)

S = (..., -13, -10, -6, -3, 0, 4, 7, 11, 14, ...).

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

A184580 = (1,2,3,5,6,...), positive terms of R;

A184581 = (4,7,11,14,...), positive terms of S;

A258833 = (1,2,4,5,7,...), - (negative terms of R);

A258834 = (0,3,6,10,...), - (nonpositive terms of S).

A184580 and A184581 partition the positive integers, and A258833 and A258834 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.
• Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.

Crossrefs

Cf. A258834 (complement), A184580, A184581.

Programs

Magma

[Ceiling((n + 1/4)*Sqrt(2)): n in [0..80]]; // Vincenzo Librandi, Jun 13 2015

Mathematica

r = Sqrt[2]; s = r/(r - 1);
Table[Ceiling[(n + 1/4) r], {n, 0, 100}] (* A258833 *)
Table[Ceiling[(n - 1/4) s], {n, 0, 100}] (* A258834 *)

PARI

for(n=0,50, print1(ceil((n + 1/4)*sqrt(2)), ", ")) \\ G. C. Greubel, Feb 08 2018

Formula

a(n) = ceiling((n + 1/4)*sqrt(2)) = floor((n + 1/4)*sqrt(2) + 1).