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A258834 Nonhomogeneous Beatty sequence: ceiling((n - 1/4)*(2 + sqrt(2)). 2
0, 3, 6, 10, 13, 17, 20, 24, 27, 30, 34, 37, 41, 44, 47, 51, 54, 58, 61, 65, 68, 71, 75, 78, 82, 85, 88, 92, 95, 99, 102, 105, 109, 112, 116, 119, 123, 126, 129, 133, 136, 140, 143, 146, 150, 153, 157, 160, 164, 167, 170, 174, 177, 181, 184, 187, 191, 194 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Complement of A248833.

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 = (1,2,4,5,6,...), - (negative terms of R);

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

A184580 and A184581 partition the positive integers, and A258833 and A248834 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.

FORMULA

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

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 *)

PROG

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

(PARI) vector(60, n, ceil((n-1/4)*(2+sqrt(2)))) \\ G. C. Greubel, Aug 19 2018

CROSSREFS

Cf. A258833 (complement), A184580, A184581.

Sequence in context: A001952 A189795 A145383 * A194028 A047280 A310054

Adjacent sequences:  A258831 A258832 A258833 * A258835 A258836 A258837

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 12 2015

STATUS

approved

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Last modified August 25 10:38 EDT 2019. Contains 326324 sequences. (Running on oeis4.)