A094077 -id:A094077 - cp's OEIS frontend

A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).

0, 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 76, 77, 79, 80, 82, 83, 84, 86, 87, 89, 90, 91, 93, 94, 96, 97, 98, 100

Offset: 0

N. J. A. Sloane

Earliest monotonic sequence greater than 0 satisfying the condition: "a(n) + 2n is not in the sequence". - Benoit Cloitre, Mar 25 2004
Also the integer part of the hypotenuse of isosceles right triangles. The real part of these numbers is irrational. For proof see Jones and Jones.
First differences are 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, ... (A006337 with a 1 in front). - Philippe Deléham, May 29 2006
It appears that the distance between the a(n)-th triangular number and the nearest square is not greater than floor(a(n)/2). - Ralf Stephan, Sep 14 2013
These are the nonnegative integers m satisfying sin(m*Pi/r)*sin((m+1)*Pi/r) <= 0, where r = sqrt(2). In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying sin(m*x)*sin((m+1)*x) <= 0, where x = Pi/r. Thus the numbers m satisfying sin(m*x)*sin((m+1)*x) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 21 2014
For n > 0: A080764(a(n)) = 1. - Reinhard Zumkeller, Jul 03 2015
From Clark Kimberling, Oct 17 2016: (Start)
We can generate A001951 and A001952 without using sqrt(2).
First write the even positive integers in a row:
  2   4   6   8   10   12   14 . . .
Then put 1 under 2 and add:
  2   4   6   8   10   12   14 . . .
  1
  3
Next, under 4, put the least positive integer that is not yet in rows 2 and 3;
it is 2; and add:
  2   4   6   8   10   12   14 . . .
  1   2
  3   6
Next, under the 6 in row 1, put the least positive integer not yet in rows 2 and 3;
it is 4, and add:
  2   4   6   8   10   12   14 . . .
  1   2   4
  3   6   10
Continue in this manner. (End)
This sequence contains an infinite number of powers of 2 (proof in Crux Mathematicorum link). See A103341. - Bernard Schott, Mar 08 2019
The terms of this sequence generate the multiplicative group of positive rational numbers (observation by Stephen M. Gagola, Jr.; see References). - Allen Stenger, Aug 05 2023
a(n) is also the number of distinct straight cylinders with integer radius and height having the same surface as a sphere with radius n. - Felix Huber, Sep 20 2024
Let P(x,y) be the condition x^2 + y^2 <= n^2, then 4*a(n) is the number of integer points (x,y) such that P(x,y) is true and at least one of P(x+1,y), P(x-1,y), P(x,y+1), P(x,y-1) is false. See LINKS for examples. - Bob de Boisvilliers, May 14 2025

Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5; MSRI Publications, Vol. 70 (2017), pages 101-153.
Stephen M. Gagola Jr., Solution of Problem 12282, Am. Math. Monthly, 130 (2023), pp. 682-683.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 77.
Gareth A. Jones and J. Mary Jones, Elementary Number Theory, Springer, 1998; pp. 221-222.
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).
Roland Sprague, Recreations in Mathematics, Blackie and Son, (1963).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), Entry sqrt(2), p. 18.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
L. Carlitz, Richard Scoville and Verner E. Hoggatt, Jr., Pellian representatives, Fib. Quart., 10 (1972), 449-488.
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190
Bob de Boisvilliers, ASCII circles' boundary pixels count examples from n=0 to n=10
Ed Doolittle, Problem 19, 26th I.M.O. Finland proposed by Romania, Crux Mathematicorum, p. 70, Vol. 14, Mar. 88.
Aviezri S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=2).
Aviezri S. Fraenkel, On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications, Discrete Mathematics 224 (2000), no. 1-3, pp. 273-279.
Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics, Aug 27 2014; See Table 3.
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)
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Complement of A001952. Equals A001952(n) - 2*n for n>0.
Equals A003151(n) - n; a bisection of A094077.
Bisections: A022842, A342281.
Cf. A022342, A026250, A080764, A103341.
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
Partial sums: A194102.

a001951 = floor . (* sqrt 2) . fromIntegral
-- Reinhard Zumkeller, Sep 14 2014

[Floor(n*Sqrt(2)): n in [0..60]]; // Vincenzo Librandi, Oct 22 2011

[Isqrt(2*n^2):n in[0..60]]; // Jason Kimberley, Oct 28 2016

a:=n->floor(n*sqrt(2)): seq(a(n),n=0..80); # Muniru A Asiru, Mar 09 2019

Floor[Range[0, 72] Sqrt[2]] (* Robert G. Wilson v, Oct 17 2012 *)

makelist(floor(n*sqrt(2)), n, 0, 100); /* Martin Ettl, Oct 17 2012 */

f(n) = for(j=1,n,print1(floor(sqrt(2*j^2))","))

a(n)=sqrtint(2*n^2) \\ Charles R Greathouse IV, Oct 19 2016

from sympy import integer_nthroot
def A001951(n): return integer_nthroot(2*n**2,2)[0] # Chai Wah Wu, Mar 16 2021

a(n) = A000196(A001105(n)). - Jason Kimberley, Oct 26 2016
a(n) = floor(csc(1/(sqrt(2)*n))) for n > 0, since sqrt(2)*n < csc(1/(sqrt(2)*n)) < sqrt(2)*n + 1/(3*sqrt(2)*n) < floor(sqrt(2)*n) + 1 for n > 0. - Jianing Song, Sep 07 2021
a(n) = A194102(n) - A194102(n-1) for n > 0. - M. F. Hasler, Apr 23 2022

More terms from David W. Wilson, Sep 20 2000

A001952 A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).

Original entry on oeis.org

3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, 54, 58, 61, 64, 68, 71, 75, 78, 81, 85, 88, 92, 95, 99, 102, 105, 109, 112, 116, 119, 122, 126, 129, 133, 136, 139, 143, 146, 150, 153, 157, 160, 163, 167, 170, 174, 177, 180, 184, 187, 191, 194, 198

Offset: 1

N. J. A. Sloane

It appears that the distance between the a(n)-th triangular number and the nearest square is greater than floor(a(n)/2). - Ralf Stephan, Sep 14 2013

A080764(a(n)) = 0. - Reinhard Zumkeller, Jul 03 2015

Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5; MSRI Publications, Vol. 70 (2017), pages 101-153.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 77.
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 = 1..10000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr. Pellian representatives, Fibonacci Quarterly, 10, issue 5, 1972, 449-488.
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8
A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=2).
Aviezri S. Fraenkel, On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications, Discrete Mathematics 224 (2000), pp. 273-279.
Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics, 179 (2014), 28-43. See Table 3.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Complement of A001951; equals A001951(n)+2*n.
A bisection of A094077.
Bisection: A187393, A342280.
Cf. A026250, A080764.
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

a001952 = floor . (* (sqrt 2 + 2)) . fromIntegral
-- Reinhard Zumkeller, Jul 03 2015

Table[Floor[n*(2 + Sqrt[2])], {n, 60}] (* Stefan Steinerberger, Apr 15 2006 *)
Array[Floor[#(2+Sqrt[2])]&,60] (* Harvey P. Dale, Dec 01 2015 *)

a(n)=2*n+sqrtint(2*n^2) \\ Charles R Greathouse IV, Jan 05 2016

from sympy import integer_nthroot
def A001952(n): return 2*n+integer_nthroot(2*n**2,2)[0] # Chai Wah Wu, Mar 16 2021

A072061 [t], 1+[t], [2t], 2+[2t], [3t], 3+[3t], ..., where t=tau = (1+sqrt(5))/2 and []=floor.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 10, 8, 13, 9, 15, 11, 18, 12, 20, 14, 23, 16, 26, 17, 28, 19, 31, 21, 34, 22, 36, 24, 39, 25, 41, 27, 44, 29, 47, 30, 49, 32, 52, 33, 54, 35, 57, 37, 60, 38, 62, 40, 65, 42, 68, 43, 70, 45, 73, 46, 75, 48, 78, 50, 81, 51, 83, 53, 86, 55, 89, 56, 91, 58, 94

Offset: 1

Clark Kimberling, Jun 11 2002, Aug 17 2007

The same sequence can be defined as follows: "a(1) = 1 and, for n>1, a(n) = a(n-1) + n/2 if n is even, otherwise a(n) = smallest positive integer which has not yet appeared in the sequence." This was originally a separate entry in the database, contributed by John W. Layman, Jul 08 2004. Antti Karttunen noticed on Jul 10 2004 that the two entries appeared to be identical. This was finally proved by Clark Kimberling, Aug 22 2007.
A permutation of the positive integers. Bisections are the lower and upper Wythoff sequences.
The consecutive pairs (1,2), (3,5), (4,7), (6,10), ... are the much-studied Wythoff pairs, arising in connection with Wythoff's game.
Conjecture: For even n, the ratio a(n)/a(n-1) is asymptotic to (1 + sqrt(5))/2 as n becomes large. (At n=3000, the ratio is 1.61804697, compared to the exact value 1.61803399.) - John W. Layman, Jul 08 2004
A more general conjecture may be stated as follows: Define {a(n)} by a(1)=1 and, for n>1, a(n) = a(n-1)+floor(kn) if n is even, else a(n)=smallest positive integer which has not yet appeared in the sequence, where k is a positive real number. Then a(2n)/a(2n-1) is asymptotic to k+sqrt(k^2+1) for large n. - John W. Layman, Jul 08 2004

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 40.

Eric Weisstein's World of Mathematics, Wythoff's game.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000201, A001950, A026272, A072062, A094077.

[n*(1+(-1)^n)/4+Floor((2*n+1-(-1)^n)*(1+Sqrt(5))/8) : n in [1..100]]; // Wesley Ivan Hurt, Apr 10 2015

A072061:=n->n*(1+(-1)^n)/4+floor((2*n+1-(-1)^n)*(1+sqrt(5))/8): seq(A072061(n), n=1..100); # Wesley Ivan Hurt, Apr 10 2015

Table[n*(1 + (-1)^n)/4 + Floor[(2 n + 1 - (-1)^n) (1 + Sqrt[5])/8], {n, 100}] (* Wesley Ivan Hurt, Apr 10 2015 *)

lista(nn) = {v = []; for (n=1, nn, v = concat(v, nt = floor(n*(1+sqrt(5))/2)); v = concat(v, n+nt);); v;} \\ Michel Marcus, Apr 14 2015

a(n) = n*(1+(-1)^n)/4+floor((2*n+1-(-1)^n)*(1+sqrt(5))/8). - Wesley Ivan Hurt, Apr 10 2015

Edited by N. J. A. Sloane, Jul 26 2008

A088610 Starting with n = 1, a(n) is the smallest squarefree number not included earlier if n is odd, else n is the smallest nonsquarefree number.

Original entry on oeis.org

1, 4, 2, 8, 3, 9, 5, 12, 6, 16, 7, 18, 10, 20, 11, 24, 13, 25, 14, 27, 15, 28, 17, 32, 19, 36, 21, 40, 22, 44, 23, 45, 26, 48, 29, 49, 30, 50, 31, 52, 33, 54, 34, 56, 35, 60, 37, 63, 38, 64, 39, 68, 41, 72, 42, 75, 43, 76, 46, 80, 47, 81, 51, 84, 53, 88, 55, 90, 57, 92, 58, 96

Offset: 1

Amarnath Murthy, Oct 16 2003

nonn

From Antti Karttunen, Jun 04 2014: (Start)
Squarefree (A005117) and nonsquarefree numbers (A013929) interleaved, the former at odd n and the latter at even n.
A243344 is a a "recursivized" variant of this permutation. Like this one, it also satisfies the given simple identity linking the parity of n with the Moebius mu-function. (End)

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers.

Inverse: A243352.
Bisections: A005117, A013929.
Similar permutations: A075378, A073846, A237056, A072061, A094077, A167151, A088609, A243344.
Cf. A000035, A008966.

With[{max = 100}, s = Select[Range[max], SquareFreeQ]; ns = Complement[Range[max], s]; Riffle[s[[1 ;; Length[ns]]], ns]] (* Amiram Eldar, Mar 04 2024 *)

(define (A088610 n) (if (even? n) (A013929 (/ n 2)) (A005117 (/ (+ 1 n) 2))))

From Antti Karttunen, Jun 04 2014: (Start)
a(2n) = A013929(n), a(2n-1) = A005117(n).
For all n, A008966(a(n)) = A000035(n), or equally, mu(a(n)) = n modulo 2, where mu is Moebius mu (A008683). (End)

More terms from Ray Chandler, Oct 18 2003

cp's OEIS Frontend

A115966 Inverse permutation to sequence A094077.

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A117149 Trajectory of 4 under map k -> A094077(k).

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A117150 Retrograde trajectory of 4 under map k -> A094077(k).

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A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).

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A001952 A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).

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A072061 [t], 1+[t], [2t], 2+[2t], [3t], 3+[3t], ..., where t=tau = (1+sqrt(5))/2 and []=floor.

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A088610 Starting with n = 1, a(n) is the smallest squarefree number not included earlier if n is odd, else n is the smallest nonsquarefree number.

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A095720 a(1)=1 and, for n>1, a(n)=a(n-1)+Floor(3n/4) if n is even, else a(n)=smallest positive integer which has not yet appeared in the sequence.