A184581 -id:A184581 - cp's OEIS frontend

A184580 a(n) = floor((n-1/4)*sqrt(2)), complement of A184581.

1, 2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 51, 53, 54, 56, 57, 59, 60, 61, 63, 64, 66, 67, 68, 70, 71, 73, 74, 76, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 92, 94, 95, 97, 98, 100, 101, 102, 104, 105, 107, 108, 109, 111, 112, 114, 115, 117, 118, 119, 121, 122, 124, 125, 126, 128, 129, 131, 132, 133, 135, 136, 138, 139, 141, 142, 143, 145, 146, 148, 149, 150, 152, 153, 155, 156, 158, 159, 160, 162, 163, 165, 166, 167, 169

Offset: 1

Clark Kimberling, Jan 17 2011

nonn

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A184581.

[Floor((n-1/4)*Sqrt(2)): n in [1..30]]; // G. C. Greubel, Jan 27 2018

r=2^(1/2); c=1/4; s=r/(r-1);
Table[Floor[n*r-c*r],{n,1,120}]  (* this sequence *)
Table[Floor[n*s+c*s],{n,1,120}]  (* A184581 *)

for(n=1, 30, print1(floor((n-1/4)*sqrt(2)), ", ")) \\ G. C. Greubel, Jan 27 2018

a(n) = floor[(n-1/4)*sqrt(2)].

Edited by Clark Kimberling, Jun 09 2015

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

Original entry on oeis.org

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

Clark Kimberling, Jun 12 2015

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.

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.

Cf. A258834 (complement), A184580, A184581.

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

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 *)

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

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

A258834 Nonhomogeneous Beatty sequence: a(n) = ceiling((n - 1/4)*(2 + sqrt(2))).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jun 12 2015

Complement of A258833.

See A258833 for more comments.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Aviezri 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.

Cf. A258833 (complement), A184580, A184581.

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

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 *)

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

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

Corrected by Michel Dekking, Sep 19 2019

cp's OEIS Frontend

A184580 a(n) = floor((n-1/4)*sqrt(2)), complement of A184581.

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

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A258834 Nonhomogeneous Beatty sequence: a(n) = ceiling((n - 1/4)*(2 + sqrt(2))).

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