A001954 -id:A001954 - cp's OEIS frontend

A003151 Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2))).

2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, 36, 38, 41, 43, 45, 48, 50, 53, 55, 57, 60, 62, 65, 67, 70, 72, 74, 77, 79, 82, 84, 86, 89, 91, 94, 96, 98, 101, 103, 106, 108, 111, 113, 115, 118, 120, 123, 125, 127, 130, 132, 135, 137, 140, 142, 144

Offset: 1

N. J. A. Sloane

Numbers with an odd number of trailing 0's in their minimal representation in terms of the positive Pell numbers (A317204). - Amiram Eldar, Mar 16 2022
From Clark Kimberling, Dec 24 2022: (Start)
This is the first of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences.
For A003151, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
(1)  u ^ v = (2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, ...) = A003151
(2)  u ^ v' = (1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, ...) = A001954
(3)  u' ^ v = (284, 287, 289, 292, 294, 296, 299, 301, 304, 306, ...) = A356135
(4)  u' ^ v' = (3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, ...) = A003152
For results of compositions instead of intersections, see A184922. (End)
The indices of the twice squares in the sequence of squares and twice squares: A028982(a(n)) = 2*n^2. - Amiram Eldar, Apr 13 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Shiri Artstein-Avidan, Aviezri S. Fraenkel and Vera T. Sós, A two-parameter family of an extension of Beatty sequences, Discrete Math., Vol. 308, No. 20 (2008), pp. 4578-4588; preprint.
Leonard Carlitz, Richard Scoville, and Verner E. Hoggatt, Jr., Pellian representatives, Fib. Quart., Vol. 10, No. 5 (1972), pp. 449-488.
Benoit Cloitre, N. J. A. Sloane, and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seq., Vol. 6 (2003), Article 03.2.2; arXiv preprint, arXiv:math/0305308 [math.NT], 2003.
Joshua N. Cooper and Alexander W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, J. Int. Seq., Vol. 16 (2013), Article 13.1.8; preprint, 2012.
R. J. Mathar, Graphical representation among sequences closely related to this one (cf. N. J. A. Sloane, "Families of Essentially Identical Sequences").
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
Index entries for sequences related to Beatty sequences.

Complement of A003152.
Equals A001951(n) + n.
Cf. A028982, A109250, A317204.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021
Bisections: A197878, A215247.
Cf. A001954, A356135, A184922, A341239.

Table[Floor[n*(1 + Sqrt[2])], {n, 1, 50}] (* G. C. Greubel, Jul 02 2017 *)

for(n=1,50, print1(floor(n*(1 + sqrt(2))), ", ")) \\ G. C. Greubel, Jul 02 2017

from math import isqrt
def A003151(n): return n+isqrt(n*n<<1) # Chai Wah Wu, Aug 03 2022

a(1) = 2; for n>1, a(n+1) = a(n)+3 if n is already in the sequence, a(n+1) = a(n)+2 otherwise.

A003152 A Beatty sequence: a(n) = floor(n*(1+1/sqrt(2))).

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 11, 13, 15, 17, 18, 20, 22, 23, 25, 27, 29, 30, 32, 34, 35, 37, 39, 40, 42, 44, 46, 47, 49, 51, 52, 54, 56, 58, 59, 61, 63, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 81, 83, 85, 87, 88, 90, 92, 93, 95, 97, 99, 100, 102, 104, 105, 107, 109, 110, 112, 114, 116

Offset: 1

N. J. A. Sloane

Numbers with an even number of trailing 0's in their minimal representation in terms of the positive Pell numbers (A317204). - Amiram Eldar, Mar 16 2022

The indices of the squares in the sequence of squares and twice squares: A028982(a(n)) = n^2. - Amiram Eldar, Apr 13 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..10000
Leonard Carlitz, Richard Scoville, and Verner E. Hoggatt, Jr., Pellian representations, Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.
Joshua N. Cooper and Alexander W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, J. Int. Seq., Vol. 16 (2013), Article 13.1.8; preprint, 2012.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
Index entries for sequences related to Beatty sequences.

Complement of A003151.
Cf. A028982, A109250, A317204.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021
Bisections: A001952, A001954.

[Floor(n*(1+1/Sqrt(2))): n in [1..70]]; // Vincenzo Librandi, Dec 26 2015

Digits := 100: t := evalf(1+sin(Pi/4)): A:= n->floor(t*n): seq(floor((t*n)),n=1..68); # Zerinvary Lajos, Mar 27 2009

Table[Floor[n (1 + 1/Sqrt[2])], {n, 70}] (* Vincenzo Librandi, Dec 26 2015 *)

a(n)=n+sqrtint(2*n^2)\2 \\ Charles R Greathouse IV, Jan 25 2022

from math import isqrt
def A003152(n): return n+isqrt(n**2>>1) # Chai Wah Wu, May 24 2025

A136119 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the term at position n + a(n).

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 71, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 95, 96, 97, 99, 100

Offset: 1

Ctibor O. Zizka, Mar 16 2008

Apparently a(n) = A001953(n-1)+1 = floor((n-1/2)*sqrt(2))+1 (confirmed for n < 20000) and a(n+1) - a(n) = A001030(n). From the definitions these conjectures are by no means obvious. Can they be proved? - Klaus Brockhaus, Apr 15 2008 [For an affirmative answer, see the Cloitre link.]

This is the s(n)-Wythoff sequence for s(n)=2n-1; see A184117 for the definition. Complement of A184119. - Clark Kimberling, Jan 09 2011

			First few steps are:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 1; delete term at position 1+a(1) = 2: 2;
1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 2; delete term at position 2+a(2) = 5: 6;
1,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 3; delete term at position 3+a(3) = 7: 9;
1,3,4,5,7,8,10,11,12,13,14,15,16,17,18,19,20,...
n = 4; delete term at position 4+a(4) = 9: 12;
1,3,4,5,7,8,10,11,13,14,15,16,17,18,19,20,...
n = 5; delete term at position 5+a(5) = 12: 16;
1,3,4,5,7,8,10,11,13,14,15,17,18,19,20,...
n = 6; delete term at position 6+a(6) = 14: 19;
1,3,4,5,7,8,10,11,13,14,15,17,18,20,...

B. Cloitre, The golden sieve, preprint 2008

Alois P. Heinz, Table of n, a(n) for n = 1..1000
D. X. Charles, Sieve Methods, July 2000, University of Wisconsin.
Benoit Cloitre, On the proof of Klaus Brockhaus's conjectures
R. Eismann, Decomposition of natural numbers into weight X level + jump and application to a new classification of prime numbers, arXiv:0711.0865 [math.NT], 2007-2010.
M. C. Wunderlich, A general class of sieve generated sequences, Acta Arithmetica XVI,1969, pp.41-56.
Index entries for sequences generated by sieves

Cf. A000027, A001953 (floor((n+1/2)*sqrt(2))), A001030 (fixed under 1 -> 21, 2 -> 211), A136110, A137292.
Cf. A000959, A099267.
Cf. A242535.
Cf. A000217 (T).

import Data.List (delete)
a136119 n = a136119_list !! (n-1)
a136119_list = f [1..] where
   f zs@(y:xs) = y : f (delete (zs !! y) xs)
-- Reinhard Zumkeller, May 17 2014

[Ceiling((n-1/2)*Sqrt(2)): n in [1..100]]; // Vincenzo Librandi, Jul 01 2019

f[0] = Range[100]; f[n_] := f[n] = Module[{pos = n + f[n-1][[n]]}, If[pos > Length[f[n-1]], f[n-1], Delete[f[n-1], pos]]]; f[1]; f[n = 2]; While[f[n] != f[n-1], n++]; f[n] (* Jean-François Alcover, May 08 2019 *)
T[n_] := n (n + 1)/2; Table[1 + 2 Sqrt[T[n-1]] , {n, 1, 71}] // Floor (* Ralf Steiner, Oct 23 2019 *)

apply( {A136119(n)=sqrtint(n*(n-1)*2)+1}, [1..99]) \\ M. F. Hasler, Jul 04 2022

a(n) = ceiling((n-1/2)*sqrt(2)). This can be proved in the same way as the formula given for A099267. There are some generalizations. For instance, it is possible to consider "a(n)+K*n" instead of "a(n)+n" for deleting terms where K=0,1,2,... is fixed. The constant involved in the Beatty sequence for the sequence of deleted terms then depends on K and equals (K + 1 + sqrt((K+1)^2 + 4))/2. K=0 is related to A099267. 1+A001954 is the complement sequence of this sequence A136119. - Benoit Cloitre, Apr 18 2008
a(n) = floor(1 + 2*sqrt(T(n-1))), with triangular numbers T(). - Ralf Steiner, Oct 23 2019
Lim_{n->inf}(a(n)/(n - 1)) = sqrt(2), with {a(n)/(n - 1)} decreasing. - Ralf Steiner, Oct 24 2019

Edited and extended by Klaus Brockhaus, Apr 15 2008

An incorrect g.f. removed by Alois P. Heinz, Dec 14 2012

A001953 a(n) = floor((n + 1/2) * sqrt(2)).

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 53, 54, 55, 57, 58, 60, 61, 62, 64, 65, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 88, 89, 91, 92, 94, 95

Offset: 0

N. J. A. Sloane

nonn

Let s(n) = zeta(3) - Sum_{k = 1..n} 1/k^3. Conjecture: for n >= 1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1's, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimberling, Oct 05 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Complement of A001954.

Cf. A000217 (T), A136119, A001108.

[Floor((2*n+1)/Sqrt(2)): n in [0..100]]; // G. C. Greubel, Nov 14 2019

seq( floor((2*n+1)/sqrt(2)), n=0..100); # G. C. Greubel, Nov 14 2019

Table[Floor[(n + 1/2) Sqrt[2]], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

PARI
```
a(n)=floor((n+1/2)*sqrt(2))
```

a(n)={sqrtint(2*n*(n+1))} \\ Andrew Howroyd, Oct 24 2019

[floor((2*n+1)/sqrt(2)) for n in (0..100)] # G. C. Greubel, Nov 14 2019

From Ralf Steiner, Oct 23 2019: (Start)
a(n) = floor(2*sqrt(A000217(n))).
a(n) = A136119(n + 1) - 1.
a(n + 1) - a(n) is in {1,2}.
a(n + 3) - a(n) is in {4,5}. (End)

More terms from Michael Somos, Apr 26 2000.

A049473 Nearest integer to n/sqrt(2).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47

Offset: 0

N. J. A. Sloane

nonn

a(n) = floor(n*sqrt(2)) - floor(n/sqrt(2)). Indeed, the equation {(nearest integer to n/r) = floor(nr) - floor(n/r) for all n>=0} has exactly two solutions: sqrt(2) and -sqrt(2). - Clark Kimberling, Dec 18 2003

Let s(n) = zeta(3) - Sum_{k=1..n} 1/k^3. Conjecture: for n >=1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimberling, Oct 05 2014

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A091087.

[0] cat [Round(n/Sqrt(2)): n in [1..100]]; // G. C. Greubel, Jan 27 2018

Round[Range[0,70]/Sqrt[2]] (* Harvey P. Dale, Feb 17 2015 *)

a(n)=round(n/sqrt(2)) \\ Charles R Greathouse IV, Sep 02 2015

A001963 Winning positions in the u-pile of the 4-Wythoff game with i=1.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79, 80, 81, 83

Offset: 0

N. J. A. Sloane

See Connell (1959) for further information.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190

Cf. A001954, A001958, A001959, A001964-A001968.

Table[Floor[(n + 1/4)*(Sqrt[5] - 1)], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

a(n) = floor( (n+1/4)*(sqrt(5)-1) ). - R. J. Mathar, Feb 14 2011

Edited by Hugo Pfoertner, Dec 27 2021

A001960 a(n) = floor((n+2/3)*(5+sqrt(13))/2); v-pile positions in the 3-Wythoff game.

Original entry on oeis.org

2, 7, 11, 15, 20, 24, 28, 32, 37, 41, 45, 50, 54, 58, 63, 67, 71, 76, 80, 84, 88, 93, 97, 101, 106, 110, 114, 119, 123, 127, 131, 136, 140, 144, 149, 153, 157, 162, 166, 170, 174, 179, 183, 187, 192, 196, 200, 205, 209, 213, 218, 222, 226, 230, 235, 239, 243, 248

Offset: 0

N. J. A. Sloane

nonn

3-Wythoff game, i=2, the v-pile positions in the Connell terminology. - R. J. Mathar, Feb 14 2011

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190.

Complement of A001957.

Cf. A001954, A001958, A001959, A001963-A001968.

Table[Floor[(n + 2/3)*(5 + Sqrt[13])/2], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

a(n) = floor((n+2/3)*(5+sqrt(13))/2). - R. J. Mathar, Feb 14 2011

New name from Hugo Pfoertner, Dec 27 2021

A001959 u-pile numbers for the 3-Wythoff game with i=2.

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 82, 84, 85, 86, 88

Offset: 0

N. J. A. Sloane

See Connell (1959) for further information.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190

Cf. A001954, A001958, A001960, A001963-A001968.

Table[Floor[(n + 2/3)*(Sqrt[13] - 1)/2], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

a(n) = floor( (n+2/3)*(sqrt(13)-1)/2 ). - R. J. Mathar, Feb 14 2011

Edited by Hugo Pfoertner, Dec 27 2021

A383032 Exponent of the highest power of 2 dividing the n-th number that is either a square or twice a square.

Original entry on oeis.org

0, 1, 2, 3, 0, 4, 1, 0, 5, 2, 0, 1, 6, 3, 0, 1, 2, 0, 7, 4, 1, 0, 2, 3, 0, 1, 8, 5, 0, 2, 1, 0, 3, 4, 0, 1, 2, 9, 0, 6, 1, 0, 3, 2, 1, 0, 4, 5, 0, 1, 2, 0, 3, 10, 1, 0, 7, 2, 0, 1, 4, 3, 0, 2, 1, 0, 5, 6, 0, 1, 2, 3, 0, 1, 4, 0, 11, 2, 1, 0, 8, 3, 0, 1, 2, 5, 0

Offset: 1

Amiram Eldar, Apr 13 2025

The first position of k in this sequence, for k >= 0, is A001521(k+1).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001521, A001954, A003151, A003152, A007814, A014176, A028982, A215247, A342280.

With[{m = 3000}, IntegerExponent[Union[Join[Range[Floor[Sqrt[m]]]^2, 2*Range[Floor[Sqrt[m/2]]]^2]], 2]]

lista(nn) = apply(x->valuation(x,2), vecsort(concat(vector(sqrtint(nn\1), i, i^2), vector(sqrtint(nn\2), i, 2*i^2)))); \\ Michel Marcus, Apr 13 2025

a(n) = A007814(A028982(n)).
a(A001954(n)) = 0 for n >= 0.
a(A215247(n)) = 1 for n >= 1.
a(A342280(n)) = 2 for n >= 0.
Sum_{k=1..n} a(k) ~ (1 + sqrt(2)) * n.

A001957 u-pile positions in the 3-Wythoff game with i=1.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87

Offset: 0

N. J. A. Sloane

See Connell (1959) for further information.

The complement is A001960. - Omar E. Pol, Jan 06 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190.

Cf. A001954, A001958-A001960, A001963-A001968.

Table[Floor[(n + 1/3)*(Sqrt[13] - 1)/2], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

a(n) = floor((n+1/3)*(sqrt(13)-1)/2). - R. J. Mathar, Feb 14 2011

Edited by N. J. A. Sloane, Dec 27 2021

cp's OEIS Frontend

A003151 Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2))).

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A003152 A Beatty sequence: a(n) = floor(n*(1+1/sqrt(2))).

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A136119 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the term at position n + a(n).

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A001953 a(n) = floor((n + 1/2) * sqrt(2)).

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A049473 Nearest integer to n/sqrt(2).

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A001963 Winning positions in the u-pile of the 4-Wythoff game with i=1.

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