A115004 -id:A115004 - cp's OEIS frontend

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

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

A127473 a(n) = phi(n)^2.

Original entry on oeis.org

1, 1, 4, 4, 16, 4, 36, 16, 36, 16, 100, 16, 144, 36, 64, 64, 256, 36, 324, 64, 144, 100, 484, 64, 400, 144, 324, 144, 784, 64, 900, 256, 400, 256, 576, 144, 1296, 324, 576, 256, 1600, 144, 1764, 400, 576, 484, 2116, 256, 1764, 400, 1024, 576, 2704, 324, 1600

Offset: 1

Gary W. Adamson, Jan 15 2007

Number of maps of the form j |--> m*j + d with gcd(m, n) = 1 and gcd(d, n) = 1 from [1, 2, ..., n] to itself. - Joerg Arndt, Aug 29 2014
Right border of A127474.
Equals the Mobius transform (A054525) of A029939. - Gary W. Adamson, Aug 20 2008
From Jianing Song, Apr 14 2019: (Start)
a(n) is the number of solutions to gcd(xy, n) = 1 with x, y in [0, n-1].
Let Z_n be the ring of integers modulo n, then a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - x) (or equivalently, Z_n[x]/(x^2 + x)) with discriminant d = 1 (that is, a(n) is the size of the group G(n) = (Z_n[x]/(x^2 - x))*). Actually, G(n) is isomorphic to (Z_n)* X (Z_n)*. (End)

			a(5) = 16 since phi(5) = 4.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Cf. A000010, A057434, A065464, A109695, A127474, A358714, A361148, A008683.

Similar sequences: A082953 (size of (Z_n[x]/(x^2 - 1))*, d = 4), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).

[(EulerPhi(n))^2: n in [1..180]]; // Vincenzo Librandi, Apr 04 2011

A127473 := proc(n) numtheory[phi](n)^2 ; end proc:
seq(A127473(n),n=1..40) ; # R. J. Mathar, Apr 04 2011

Table[EulerPhi[n]^2,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

a(n) = eulerphi(n)^2; \\ Michel Marcus, Oct 16 2018

a(n) = A000010(n)^2.
Multiplicative with a(p^e) = (p-1)^2*p^(2e-2), e >= 1. Dirichlet g.f. zeta(s-2)*Product_{primes p} (1 - 2/p^(s-1) + 1/p^s). - R. J. Mathar, Apr 04 2011
Sum_{k>=1} 1/a(k) = A109695. - Vaclav Kotesovec, Sep 20 2020
Sum_{k>=1} (-1)^k/a(k) = (1/7) * A109695. - Amiram Eldar, Nov 11 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime}(1 - (2*p-1)/p^3) = A065464 / 3 = 0.142749... . - Amiram Eldar, Oct 25 2022
a(n) = Sum_{d|n} mu(n/d)*phi(n*d). - Ridouane Oudra, Jul 23 2025

A114043 Take an n X n square grid of points in the plane; a(n) = number of ways to divide the points into two sets using a straight line.

Original entry on oeis.org

1, 7, 29, 87, 201, 419, 749, 1283, 2041, 3107, 4493, 6395, 8745, 11823, 15557, 20075, 25457, 32087, 39725, 48935, 59457, 71555, 85253, 101251, 119041, 139351, 161933, 187255, 215137, 246691, 280917, 319347, 361329, 407303

Offset: 1

Ugo Merlone (merlone(AT)econ.unito.it) and N. J. A. Sloane, Feb 22 2006

Also, half of the number of two-dimensional threshold functions (A114146).
The line may not pass through any point. This is the "labeled" version - rotations and reflections are not taken into account (cf. A116696).
The number of ways to divide a (2n) X (2n) grid into two sets of equal size is given by 2*A099957(n). - David Applegate, Feb 23 2006
All terms are odd: the line that misses the grid contributes 1 to the total and all other lines contribute 2, 4 or 8, so the total must be odd.
What can be said about the 3-D generalization? - Max Alekseyev, Feb 27 2006

			Examples: the two sets are indicated by X's and o's.
a(2) = 7:
XX oX Xo XX XX oo oX
XX XX XX Xo oX XX oX
--------------------
a(3) = 29:
XXX oXX ooX ooo ooX ooo
XXX XXX XXX XXX oXX oXX
XXX XXX XXX XXX XXX XXX
-1- -4- -8- -4- -4- -8- Total = 29
--------------------
a(4)= 87:
XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX
XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX
XXXX XXXX XXXX XXXX XXXX XXXo XXXo XXXo XXoo XXoo
XXXX XXXo XXoo Xooo oooo XXoo Xooo oooo Xooo oooo
--1- --4- --8- --8- --4- --4- --8- --8- --8- --8-
XXXX XXXX XXXX XXXX XXXX
XXXo XXXX XXXX XXXo XXXo
XXoo Xooo oooo Xooo XXoo
Xooo oooo oooo oooo oooo
--4- --8- --2- --4- --8- Total = 87.
--------------------

T. D. Noe, Table of n, a(n) for n = 1..1000
Max A. Alekseyev. On the number of two-dimensional threshold functions, arXiv:math/0602511 [math.CO], 2006-2010; doi:10.1137/090750184, SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631.
M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Eq. (11).
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A114499, A115004, A115005, A116696 (unlabeled case), A114531, A114146.
Cf. A099957.
The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n. - N. J. A. Sloane, Feb 04 2020

a[n_] := 2*Sum[(n - i)*(n - j)*Boole[CoprimeQ[i, j]], {i, 1, n - 1}, {j, 1, n - 1}] + 2*n^2 - 2*n + 1; Array[a, 40] (* Jean-François Alcover, Apr 25 2016, after Max Alekseyev *)

from sympy import totient
def A114043(n): return 4*n**2-6*n+3 + 2*sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n)) # Chai Wah Wu, Aug 15 2021

Let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)*(n+1-j); then a(n+1) = 2*(n^2 + n + V(n,n)) + 1. - Max Alekseyev, Feb 22 2006
a(n) ~ (3/Pi^2) * n^4. - Max Alekseyev, Feb 22 2006
a(n) = A141255(n) + 1. - T. D. Noe, Jun 17 2008
a(n) = 4*n^2 - 6*n + 3 + 2*Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i). - Chai Wah Wu, Aug 15 2021

More terms from Max Alekseyev, Feb 22 2006

A141255 Total number of line segments between points visible to each other in a square n X n lattice.

Original entry on oeis.org

0, 6, 28, 86, 200, 418, 748, 1282, 2040, 3106, 4492, 6394, 8744, 11822, 15556, 20074, 25456, 32086, 39724, 48934, 59456, 71554, 85252, 101250, 119040, 139350, 161932, 187254, 215136, 246690, 280916, 319346, 361328, 407302, 457180, 511714, 570232

Offset: 1

T. D. Noe, Jun 17 2008

nonn

A line segment joins points (a,b) and (c,d) if the points are distinct and gcd(c-a,d-b)=1.

			The 2 x 2 square lattice has a total of 6 line segments: 2 vertical, 2 horizontal and 2 diagonal.

D. M. Acketa, J. D. Zunic: On the number of linear partitions of the (m,n)-grid. Inform. Process. Lett., 38 (3) (1991), 163-168. See Table A.1.
Jovisa Zunic, Note on the number of two-dimensional threshold functions, SIAM J. Discrete Math. Vol. 25 (2011), No. 3, pp. 1266-1268. See Eq. (1.2).

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points [From _Seppo Mustonen_, May 13 2010]
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points [Local copy]
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A141224.

The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n. - N. J. A. Sloane, Feb 04 2020

Table[cnt=0; Do[If[GCD[c-a,d-b]<2, cnt++ ], {a,n}, {b,n}, {c,n}, {d,n}]; (cnt-n^2)/2, {n,20}]
(* This recursive code is much more efficient. *)
a[n_]:=a[n]=If[n<=1,0,2*a1[n]-a[n-1]+R1[n]]
a1[n_]:=a1[n]=If[n<=1,0,2*a[n-1]-a1[n-1]+R2[n]]
R1[n_]:=R1[n]=If[n<=1,0,R1[n-1]+4*EulerPhi[n-1]]
R2[n_]:=(n-1)*EulerPhi[n-1]
Table[a[n],{n,1,37}]
(* Seppo Mustonen, May 13 2010 *)
a[n_]:=2 Sum[(n-i) (n-j) Boole[CoprimeQ[i,j]], {i,1,n-1}, {j,1,n-1}] + 2 n^2 - 2 n; Array[a, 40] (* Vincenzo Librandi, Feb 05 2020 *)

from sympy import totient
def A141255(n): return 2*(n-1)*(2*n-1) + 2*sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n)) # Chai Wah Wu, Aug 16 2021

a(n) = A114043(n) - 1.

a(n) = 2*(n-1)*(2n-1) + 2*Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i). - Chai Wah Wu, Aug 16 2021

A159684 Sturmian word: limit S(infinity) where S(0) = 0, S(1) = 0,1 and for n>=1, S(n+1) = S(n)S(n)S(n-1).

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0

Offset: 0

Philippe Deléham, Apr 19 2009

Fixed point of morphism 0 -> 0,1; 1 -> 0,1,0.
This sequence corresponds to the case k = 1 of the Sturmian word S_k(infinity) as defined in A080764. See A171588 for the case k = 2. - Peter Bala, Nov 22 2013
This sequence is the {1->01}-transform of the Sturmian word A080764.  - Clark Kimberling, May 17 2017
Also, sequence (1 if x/sqrt(2) is integer, 0 else) as x runs over the elements of N U N*sqrt(2) in increasing order, N = {0, 1, 2, ...}. See A144612 for the sqrt(3) analog. - M. F. Hasler, Feb 06 2025

			0 -> 0,1 -> 0,1,0,1,0 -> 0,1,0,1,0,0,1,0,1,0,0,1 ->...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, and P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
Jeffrey Shallit, Characteristic words as fixed points of homomorphisms, University of Waterloo Technical Report CS-91-72, 1991. See Example 1.
Jeffrey Shallit, Characteristic words as fixed points of homomorphisms. See Example 1. [Cached copy, with permission]
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Vincent Van Dongen, Tileset As: 3 squares with local rules for non-periodic tiling, arXiv:2503.11682 [math.GM], 2025. See p. 5.
Wikipedia, Sturmian word

See A188037 for another version of this sequence. - N. J. A. Sloane, Mar 22 2011
Cf. A000129, A001333, A005614, A080764, A119812, A171588, A221150, A221151, A221152, A096270.
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

a159684 n = a159684_list !! n
a159684_list = 0 : concat (iterate (concatMap s) [1])
   where s 0 = [0,1]; s 1 = [0,1,0]
-- Reinhard Zumkeller, Oct 26 2013

Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {0, 1, 0}}] &, {1}, 6] (* Robert G. Wilson v, May 02 2009 *)
SubstitutionSystem[{0->{0,1},1->{0,1,0}},{1},{6}][[1]] (* Harvey P. Dale, Dec 25 2021 *)

def aupto(nn):
    Snm1, Sn = [0], [0, 1]
    while len(Sn) < nn+1: Snm1, Sn = Sn, Sn + Sn + Snm1
    return Sn[:nn+1]
print(aupto(104)) # Michael S. Branicky, Jul 23 2022

from math import isqrt
def A159684(n): return -isqrt(m:=(n+1)**2<<1)+isqrt(m+(n<<2)+6)-1 # Chai Wah Wu, Aug 03 2022

From Peter Bala, Nov 22 2013: (Start)
a(n) = floor((n + 2)*(sqrt(2) - 1)) - floor((n + 1)*(sqrt(2) - 1)).
If we read the sequence as the decimal constant C = 0.01010 01010 01010 10010 10010 ... then C = sum {n >= 1} 1/10^floor(n*(1 + sqrt(2))).
The real number 9*C has the simple continued fraction expansion [0; 11, 1010, 10000100, 100000000000100000, 100000000000000000000000000001000000000000, ...], the partial quotients having the form 10^Pell(n)*(1 + 10^Pell(n+1)) = 10^A001333(n+1) + 10^A000129(n) (see Adams and Davison).
A rapidly converging series for C is C = 9*sum {n >= 1} 10^Pell(2*n-1)*(1 + 10^Pell(2*n))/( (10^Pell(2*n-1) - 1)*(10^Pell(2*n+1) - 1) ): for example, the first 10 terms of the series give a rational approximation to C accurate to more than 130 million decimal places. Compare with the Fibonacci words A005614 and A221150. (End)

More terms from Robert G. Wilson v, May 02 2009

A001030 Fixed under 1 -> 21, 2 -> 211.

Original entry on oeis.org

2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2

Offset: 1

N. J. A. Sloane

If treated as the terms of a continued fraction, it converges to approximately
2.57737020881617828717350576260723346479894963737498275232531856357441\
7024804797827856956758619431996. - Peter Bertok (peter(AT)bertok.com), Nov 27 2001
There are a(n) 1's between successive 2's. - Eric Angelini, Aug 19 2008
Same sequence where 1's and 2's are exchanged: A001468. - Eric Angelini, Aug 19 2008

Midhat J. Gazale, Number: From Ahmes to Cantor, Section on 'Cleavages' in Chapter 6, Princeton University Press, Princeton, NJ 2000, pp. 203-211.
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..8119
N. G. de Bruijn, Sequences of zeros and ones generated by special production rules, Indag. Math., 43 (1981), 27-37.
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
A. Nagel, A self-defining infinite sequence, with an application to Markoff chains and probability, Math. Mag., 36 (1963), 179-183.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282).
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Length of the sequence after 'n' substitution steps is given by the terms of A000129.
Equals A004641(n) + 1.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

Following Spage's PARI program.
a001030 n = a001030_list !! (n-1)
a001030_list = [2, 1, 1, 2] ++ f [2] [2, 1, 1, 2] where
   f us vs = ws ++ f vs (vs ++ ws) where
             ws = 1 : us ++ 1 : vs
-- Reinhard Zumkeller, Aug 04 2014

('n' is the number of substitution steps to perform.) Nest[Flatten[ # /. {1 -> {2, 1}, 2 -> {2, 1, 1}}] &, {1}, n]
SubstitutionSystem[{1->{2,1},2->{2,1,1}},{2},{6}][[1]] (* Harvey P. Dale, Feb 15 2022 *)

/* Fast string concatenation method giving e.g. 5740 terms in 8 iterations */
a="2";b="2,1,1,2";print1(b);for(x=1,8,c=concat([",1,",a,",1,",b]);print1(c);a=b;b=concat(b,c)) \\ K. Spage, Oct 08 2009

from math import isqrt
def A001030(n): return [2, 1, 1, 2, 1, 2, 1, 2][n-1] if n < 9 else -isqrt(m:=(n-9)*(n-9)<<1)+isqrt(m+(n-9<<2)+2) # Chai Wah Wu, Aug 25 2022

a(n) = -1 + floor(n*(1+sqrt(2))+1/sqrt(2))-floor((n-1)*(1+sqrt(2))+1/sqrt(2)). - Benoit Cloitre, Jun 26 2004. [I don't know if this is a theorem or a conjecture. - N. J. A. Sloane, May 14 2008]

This is a theorem, following from Hofstadter's Generalized Fundamental Theorem of eta-sequences on page 10 of Eta-Lore. See also de Bruijn's paper from 1981 (hint from Benoit Cloitre). - Michel Dekking, Jan 22 2017

More terms from Peter Bertok (peter(AT)bertok.com), Nov 27 2001

A004641 Fixed under 0 -> 10, 1 -> 100.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1

Offset: 1

N. J. A. Sloane

Partial sums: A088462. - Reinhard Zumkeller, Dec 05 2009

Write w(n) = a(n) for n >= 1. Each w(n) is generated by w(i) for exactly one i <= n; let g(n) = i. Each w(i) generates a single 1, in a word (10 or 100) that starts with 1. Therefore, g(n) is the number of 1s among w(1), ..., w(n), so that g = A088462. That is, this sequence is generated by its partial sums. - Clark Kimberling, May 25 2011

T. D. Noe, Table of n, a(n) for n = 1..8119
Wieb Bosma, Michel Dekking, and Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), arXiv:1710.01498 [math.NT], 2017.
Wieb Bosma, Michel Dekking, and Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A4.
N. G. de Bruijn, Sequences of zeros and ones generated by special production rules, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 1, 27-37. Reprinted in Physics of Quasicrystals, ed. P. J. Steinhardt et al., p. 664.
C. J. Glasby, S. P. Glasby, and F. Pleijel, Worms by number, Proc. Roy. Soc. B, Proc. Biol. Sci. 275 (1647) (2008) 2071-2076.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
Index entries for characteristic functions

Equals A001030 - 1. Essentially the same as A006337 - 1 and A159684.
Characteristic function of A086377.
Cf. A081477.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

[Floor(n*(Sqrt(2) - 1) + Sqrt(1/2)) - Floor((n - 1)*(Sqrt(2) - 1) + Sqrt(1/2)): n in [0..100]]; // Vincenzo Librandi, Mar 27 2015

P(0):= (1,0): P(1):= (1,0,0):
((P~)@@6)([1]);
# in Maple 12 or earlier, comment the above line and uncomment the following:
# (curry(map,P)@@6)([1]); # Robert Israel, Mar 26 2015

Nest[ Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 0}}] &, {1}, 5] (* Robert G. Wilson v, May 25 2011 *)
SubstitutionSystem[{0->{1,0},1->{1,0,0}},{1},5]//Flatten (* Harvey P. Dale, Nov 20 2021 *)

from math import isqrt
def A004641(n): return [1, 0, 0, 1, 0, 1, 0, 1][n-1] if n < 9 else -1-isqrt(m:=(n-9)*(n-9)<<1)+isqrt(m+(n-9<<2)+2) # Chai Wah Wu, Aug 25 2022

a(n) = floor(n*(sqrt(2) - 1) + sqrt(1/2)) - floor((n - 1)*(sqrt(2) - 1) + sqrt(1/2)) (from the de Bruijn reference). - Peter J. Taylor, Mar 26 2015
From Jianing Song, Jan 02 2019: (Start)
a(n) = A001030(n) - 1.
a(n) = A006337(n-9) - 1 = A159684(n-10) for n >= 10. (End)

A022842 Beatty sequence for sqrt(8).

Original entry on oeis.org

2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 31, 33, 36, 39, 42, 45, 48, 50, 53, 56, 59, 62, 65, 67, 70, 73, 76, 79, 82, 84, 87, 90, 93, 96, 98, 101, 104, 107, 110, 113, 115, 118, 121, 124, 127, 130, 132, 135, 138, 141, 144, 147, 149, 152, 155, 158, 161, 164

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Index entries for sequences related to Beatty sequences

A bisection of A001951. Cf. A010466.

[Floor(n*Sqrt(8)): n in [1..60]]; // Vincenzo Librandi, Oct 24 2011

a:=n->floor(2*n*sqrt(2)): seq(a(n),n=1..60); # Muniru A Asiru, Sep 28 2018

Table[Floor[2*n*Sqrt[2]], {n,1,60}] (* G. C. Greubel, Sep 28 2018 *)

vector(80, n, floor(2*n*sqrt(2))) \\ G. C. Greubel, Sep 28 2018

from sympy import integer_nthroot
def A022842(n): return integer_nthroot(8*n**2,2)[0] # Chai Wah Wu, Mar 16 2021

a(n) = floor(2*n*sqrt(2)). - Michel Marcus, Oct 31 2017

Offset changed from 0 to 1 by Vincenzo Librandi, Oct 24 2011

A037225 a(n) = phi(2n+1).

Original entry on oeis.org

1, 2, 4, 6, 6, 10, 12, 8, 16, 18, 12, 22, 20, 18, 28, 30, 20, 24, 36, 24, 40, 42, 24, 46, 42, 32, 52, 40, 36, 58, 60, 36, 48, 66, 44, 70, 72, 40, 60, 78, 54, 82, 64, 56, 88, 72, 60, 72, 96, 60, 100, 102, 48, 106, 108, 72, 112, 88, 72, 96, 110, 80, 100, 126, 84, 130

Offset: 0

N. J. A. Sloane

Bisection of A000010 (cf. A062570).
From Alain Rocchelli, Jun 28 2023: (Start)
If 2*n+1 has r distinct odd prime factors, 2^r divides a(n).
Conjectures:
1) For any composite integer 2*n+1, a(n) doesn't divide 2*n.
2) For all n, a(n) is never equal to n. (End)

T. D. Noe, Table of n, a(n) for n = 0..10000
John Brillhart, J. S. Lomont, and Patrick Morton, Cyclotomic properties of the Rudin-Shapiro polynomials, J. Reine Angew. Math. 288 (1976), 37-65; see Table 2; MR0498479 (58 #16589). - From _N. J. A. Sloane_, Jun 06 2012
V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems, Problemy Peredachi Informatsii, 43 (No. 3, 2007), 39-53 (in Russian).
V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems, Problems of Information. Transmission, 43 (2007), 199-212 (translation from Russian).
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
Wikipedia, Lehmer's totient problem.

Cf. A000010, A062570, A072451, A217739.

[EulerPhi(2*n+1): n in [0..70]]; // Vincenzo Librandi, Jul 16 2019

Table[EulerPhi[2 n + 1], {n, 0, 80}] (* Vincenzo Librandi, Jul 16 2019 *)

a(n) = eulerphi(2*n+1) \\ Amiram Eldar, Nov 17 2022

Sum_{k=0..n} a(k) ~ c * n^2, where c = 8/Pi^2 = 0.810569... (A217739). - Amiram Eldar, Nov 17 2022
a(n) = 2*n iff 2*n+1 is prime, see A005097. - Alain Rocchelli, Jun 22 2023
From Peter Bala, Feb 01 2024: (Start)
Odd bisection of A000010.
a(n) = 2*A072451(n) for n >= 1.
G.f.: Sum_{n >= 1} phi(2*n+1)*x^(2*n+1) = Sum_{n >= 1} moebius(n)*x^(2*n-1)*(1 + x^(4*n-2))/(1 - x^(4*n-2))^2 = x + 2*x^3 + 4*x^5 + 6*x^7 + 6*x^9 + .... (End)

A088462 a(1)=1, a(n) = ceiling((n - a(a(n-1)))/2).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32

Offset: 1

Benoit Cloitre, Nov 12 2003

nonn

Partial sums of A004641. - Reinhard Zumkeller, Dec 05 2009

This sequence generates A004641; see comment at A004641. - Clark Kimberling, May 25 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A005206.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

[Floor((Sqrt(2)-1)*n+1/Sqrt(2)): n in [1..100]]; // Vincenzo Librandi, Jun 26 2017

Table[Floor[(Sqrt[2] - 1) n + 1 / Sqrt[2]], {n, 100}] (* Vincenzo Librandi, Jun 26 2017 *)

l=[0, 1, 1]
for n in range(3, 101): l.append(n - l[n - 1] - l[l[n - 2]])
print(l[1:]) # Indranil Ghosh, Jun 24 2017, after Altug Alkan

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

a(1) = a(2) = 1; a(n) = n - a(n-1) - a(a(n-2)) for n > 2. - Altug Alkan, Jun 24 2017

cp's OEIS Frontend

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

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A127473 a(n) = phi(n)^2.

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A114043 Take an n X n square grid of points in the plane; a(n) = number of ways to divide the points into two sets using a straight line.

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A141255 Total number of line segments between points visible to each other in a square n X n lattice.

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A159684 Sturmian word: limit S(infinity) where S(0) = 0, S(1) = 0,1 and for n>=1, S(n+1) = S(n)S(n)S(n-1).

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A001030 Fixed under 1 -> 21, 2 -> 211.

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