A057211 -id:A057211 - cp's OEIS frontend

A001614 Connell sequence: 1 odd, 2 even, 3 odd, ...

1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122

Offset: 1

N. J. A. Sloane

Next (2n-1) odd numbers alternating with next 2n even numbers. Squares (A000290(n)) occur at the A000217(n)-th entry. - Lekraj Beedassy, Aug 06 2004. - Comment corrected by Daniel Forgues, Jul 18 2009
a(t_n) = a(n(n+1)/2) = n^2 relates squares to triangular numbers. - Daniel Forgues
The natural numbers not included are A118011(n) = 4n - a(n) as n=1,2,3,... - Paul D. Hanna, Apr 10 2006
As a triangle with row sums = A069778 (1, 6, 21, 52, 105, ...): /Q 1;/Q 2, 4;/Q 5, 7, 9;/Q 10, 12, 14, 16;/Q ... . - Gary W. Adamson, Sep 01 2008
The triangle sums, see A180662 for their definitions, link the Connell sequence A001614 as a triangle with six sequences, see the crossrefs. - Johannes W. Meijer, May 20 2011
a(n) = A122797(n) + n - 1. - Reinhard Zumkeller, Feb 12 2012

			From _Omar E. Pol_, Aug 13 2013: (Start)
Written as a triangle the sequence begins:
   1;
   2,  4;
   5,  7,  9;
  10, 12, 14, 16;
  17, 19, 21, 23, 25;
  26, 28, 30, 32, 34, 36;
  37, 39, 41, 43, 45, 47, 49;
  50, 52, 54, 56, 58, 60, 62, 64;
  65, 67, 69, 71, 73, 75, 77, 79, 81;
  82, 84, 86, 88, 90, 92, 94, 96, 98, 100;
  ...
Right border gives A000290, n >= 1.
(End)

C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 276.
C. A. Pickover, The Mathematics of Oz, Chapter 39, Camb. Univ. Press UK 2002.
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..1000
Ian Connell and Andrew Korsak, Problem E1382, Amer. Math. Monthly, 67 (1960), 380.
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Gary E. Stevens, A Connell-Like Sequence, J. Integer Sequences, Vol. 1, 1998, #98.1.4.
Eric Weisstein's World of Mathematics, Connell Sequence

Cf. A117384, A118011 (complement), A118012.
Cf. A069778. - Gary W. Adamson, Sep 01 2008
From Johannes W. Meijer, May 20 2011: (Start)
Triangle columns: A002522, A117950 (n>=1), A117951 (n>=2), A117619 (n>=3), A154533 (n>=5), A000290 (n>=1), A008865 (n>=2), A028347 (n>=3), A028878 (n>=1), A028884 (n>=2), A054569 [T(2*n,n)].
Triangle sums (see the comments): A069778 (Row1), A190716 (Row2), A058187 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Fi1, Fi2, Ze1 and Ze2), A000292 (Related to Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4), A190717 (Related to Ca1, Ca2, Ze3, Ze4), A190718 (Related to Gi1 and Gi2). (End)
Cf. A014132, A057211, A023531.

a001614 n = a001614_list !! (n-1)
a001614_list = f 0 0 a057211_list where
   f c z (x:xs) = z' : f x z' xs where z' = z + 1 + 0 ^ abs (x - c)
-- Reinhard Zumkeller, Dec 30 2011

[2*n-Round(Sqrt(2*n)): n in [1..80]]; // Vincenzo Librandi, Apr 17 2015

A001614:=proc(n): 2*n - floor((1+sqrt(8*n-7))/2) end: seq(A001614(n),n=1..67); # Johannes W. Meijer, May 20 2011

lst={};i=0;For[j=1, j<=4!, a=i+1;b=j;k=0;For[i=a, i<=9!, k++;AppendTo[lst, i];If[k>=b, Break[]];i=i+2];j++ ];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
row[n_] := 2*Range[n+1]+n^2-1; Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)

a(n)=2*n - round(sqrt(2*n)) \\ Charles R Greathouse IV, Apr 20 2015

from math import isqrt
def A001614(n): return (m:=n<<1)-(k:=isqrt(m))-int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

a(n) = 2*n - floor( (1+ sqrt(8*n-7))/2 ).
a(n) = A005843(n) - A002024(n). - Lekraj Beedassy, Aug 06 2004
a(n) = A118012(A118011(n)). A117384( a(n) ) = n; A117384( 4*n - a(n) ) = n. - Paul D. Hanna, Apr 10 2006
a(1) = 1; then a(n) = a(n-1)+1 if a(n-1) is a square, a(n) = a(n-1)+2 otherwise. For example, a(21)=36 is a square therefore a(22)=36+1=37 which is not a square so a(23)=37+2=39 ... - Benoit Cloitre, Feb 07 2007
T(n,k) = (n-1)^2 + 2*k - 1. - Omar E. Pol, Aug 13 2013
a(n)^2 = a(n*(n+1)/2). - Ivan N. Ianakiev, Aug 15 2013
Let the sequence be written in the form of the triangle in the EXAMPLE section below and let a(n) and a(n+1) belong to the same row of the triangle. Then a(n)*a(n+1) + 1 = a(A000217(A118011(n))) = A000290(A118011(n)). - Ivan N. Ianakiev, Aug 16 2013
a(n) = 2*n-round(sqrt(2*n)). - Gerald Hillier, Apr 15 2015
From Robert Israel, Apr 20 2015 (Start):
G.f.: 2*x/(1-x)^2 - (x/(1-x))*Sum_{n>=0} x^(n*(n+1)/2) = 2*x/(1-x)^2 - (Theta2(0,x^(1/2)))*x^(7/8)/(2*(1-x)) where Theta2 is a Jacobi theta function.
a(n) = 2*n-1 - Sum_{i=0..n-2} A023531(i).  (End)
a(n) = 3*n-A014132(n). - Chai Wah Wu, Oct 19 2024

More terms from Larry Reeves (larryr(AT)acm.org), Mar 16 2001

A193356 If n is even then 0, otherwise n.

Original entry on oeis.org

1, 0, 3, 0, 5, 0, 7, 0, 9, 0, 11, 0, 13, 0, 15, 0, 17, 0, 19, 0, 21, 0, 23, 0, 25, 0, 27, 0, 29, 0, 31, 0, 33, 0, 35, 0, 37, 0, 39, 0, 41, 0, 43, 0, 45, 0, 47, 0, 49, 0, 51, 0, 53, 0, 55, 0, 57, 0, 59, 0, 61, 0, 63, 0, 65, 0, 67, 0, 69, 0, 71, 0, 73, 0, 75

Offset: 1

José María Grau Ribas, Jul 24 2011

Multiplicative with a(2^e)=0 if e>0 and a(p^e)=p^e for odd primes p. - R. J. Mathar, Aug 01 2011
A005408 and A000004 interleaved (the usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception). - Omar E. Pol, Feb 02 2013
Row sums of A057211. - Omar E. Pol, Mar 05 2014
Column k=2 of triangle A196020. - Omar E. Pol, Aug 07 2015
a(n) is the determinant of the (n+2) X (n+2) circulant matrix with the first row [0,0,1,1,...,1]. This matrix is closely linked with the famous ménage problem (see also comments of Vladimir Shevelev in sequence A000179). Namely it defines the class of permutations p of 1,2,...,n+2 such that p(i)<>i and p(i)<>i+1 for i=1,2,...,n+1, and p(n+2)<>1,n+2. And a(n) is also the difference between the number of even and odd such permutations. - Dmitry Efimov, Feb 02 2016

Franz Lemmermeyer, Reciprocity Laws. From Euler to Eisenstein, Springer, 2000, p. 237, eq. (8.5).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. Kravvaritis, Determinant evaluations for binary circulant matrices, Special Matrices, V2(1) (2014), 187-199.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000004, A000010, A005408, A027656, A042948, A057211, A196020.

I:=[1,0,3,0]; [n le 4 select I[n] else 2*Self(n-2)-Self(n-4): n in [1..80]]; // Vincenzo Librandi, Feb 24 2014

A193356:=n->(1-(-1)^n)*n/2: seq(A193356(n), n=1..100); # Wesley Ivan Hurt, Aug 07 2015

Mathematica
```
Table[PowerMod[n,n,2*n], {n,200}]
```

a(n)=if(n%2,n) \\ Charles R Greathouse IV, Jul 24 2011

A193356 = [i if i%2 else 0 for i in range(1, 101)] # Jwalin Bhatt, Jun 17 2025

a(n) = n^k mod 2n, for any k>=2, also for k=n. [extended by Wolfdieter Lang, Dec 21 2011]
Dirichlet g.f.: (1-2^(1-s))*zeta(s-1). - R. J. Mathar, Aug 01 2011
G.f.: x*(1+x^2)/(1-x^2)^2. - Philippe Deléham, Feb 13 2012
a(n) = A027656(A042948(n-1)) = (1-(-1)^n)*n/2. - Bruno Berselli, Feb 19 2012
a(n) = n * (n mod 2). - Wesley Ivan Hurt, Jun 29 2013
G.f.: Sum_{n >= 1} A000010(n)*x^n/(1 + x^n). - Mircea Merca, Feb 22 2014
a(n) = 2*a(n-2)-a(n-4), for n>4. - Wesley Ivan Hurt, Aug 07 2015
E.g.f.: x*cosh(x). - Robert Israel, Feb 03 2016
a(n) = Product_{k=1..floor(n/2)}(sin(2*Pi*k/n))^2, for n >= 1 (with the empty product put to 1). Trivial for even n from the factor 0 for k = n/2. For odd n see, e.g., the Lemmermeyer reference, eq. (8.5) on p. 237. - Wolfdieter Lang, Aug 29 2016
a(n) = Sum_{k=1..n} (-1)^((n-k)*k). - Rick L. Shepherd, Sep 18 2020
a(n) = Sum_{k=1..n} (-1)^(1+gcd(k,n)) = Sum_{d | n} (-1)^(d+1)*phi(n/d), where phi(n) = A000010(n). - Peter Bala, Jan 14 2024
Dirichlet g.f.: DirichletLambda(s-1). - Michael Shamos, Jun 13 2025

A060734 Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are read by antidiagonals downwards.

Original entry on oeis.org

1, 4, 2, 9, 3, 5, 16, 8, 6, 10, 25, 15, 7, 11, 17, 36, 24, 14, 12, 18, 26, 49, 35, 23, 13, 19, 27, 37, 64, 48, 34, 22, 20, 28, 38, 50, 81, 63, 47, 33, 21, 29, 39, 51, 65, 100, 80, 62, 46, 32, 30, 40, 52, 66, 82, 121, 99, 79, 61, 45, 31, 41, 53, 67, 83, 101

Offset: 1

Frank Ellermann, Apr 23 2001

A simple permutation of natural numbers.
Parity of the sequence is given by A057211 (n-th run has length n). - Jeremy Gardiner, Dec 26 2008
The square with corners T(1,1)=1 and T(n,n)=n^2-n+1 is occupied by the numbers 1,2,...,n^2. - Clark Kimberling, Feb 01 2011
a(n) is pairing function - function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} - the set of integer positive numbers. - Boris Putievskiy, Dec 17 2012

			Northwest corner:
.1  4  9 16 ..  => a(1) =  1
.2  3  8 15 ..  => a(2) =  4, a(3) = 2
.5  6  7 14 ..  => a(4) =  9, a(5) = 3, a(6) = 5
10 11 12 13 ..  => a(7) = 16, a(8) = 8, a(9) = 6, a(10)=10

Alois P. Heinz, Rows n = 1..141 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060736. Inverse: A064790.

Cf. A185725, A185726, A185728.

T:= (n,k)-> `if`(n<=k, k^2-n+1, (n-1)^2+k):
seq(seq(T(n, d-n), n=1..d-1), d=2..15);

f[n_, k_]:=k^2-n+1/; k>=n;
f[n_, k_]:=(n-1)^2+k/; kClark Kimberling, Feb 01 2011 *)

T(n,k) = (n-1)^2+k, T(k, n)=n^2+1-k, 1 <= k <= n.
From Clark Kimberling, Feb 01 2011: (Start)
T(1,k) = k^2 (A000290).
T(n,n) = n^2-n+1 (A002061).
T(n,1) = (n-1)^2+1 (A002522). (End)

Corrected by Jeremy Gardiner, Dec 26 2008

A007606 Take 1, skip 2, take 3, etc.

Original entry on oeis.org

1, 4, 5, 6, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 37, 38, 39, 40, 41, 42, 43, 44, 45, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 137, 138

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

List the natural numbers: 1, 2, 3, 4, 5, 6, 7, ... . Keep the first number (1), delete the next two numbers (2, 3), keep the next three numbers (4, 5, 6), delete the next four numbers (7, 8, 9, 10) and so on.
a(A000290(n)) = A000384(n). - Reinhard Zumkeller, Feb 12 2011
A057211(a(n)) = 1. - Reinhard Zumkeller, Dec 30 2011
Numbers k with the property that the smallest Dyck path of the symmetric representation of sigma(k) has a central valley. (Cf. A237593.) - Omar E. Pol, Aug 28 2018
Union of nonzero terms of A000384 and A317304. - Omar E. Pol, Aug 29 2018
The values of k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class r' mod m' (with r' in {1,...,m'}) iff mA360418. - James Propp, Feb 10 2023

			From _Omar E. Pol_, Aug 29 2018: (Start)
Written as an irregular triangle in which the row lengths are the odd numbers the sequence begins:
    1;
    4,   5,   6;
   11,  12,  13,  14,  15;
   22,  23,  24,  25,  26,  27,  28;
   37,  38,  39,  40,  41,  42,  43,  44,  45;
   56,  57,  58,  59,  60,  61,  62 , 63,  64,  65,  66;
   79,  80,  81,  82 , 83,  84,  85,  86,  87,  88,  89,  90,  91;
  106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120;
...
Row sums give A005917.
Column 1 gives A084849.
Column 2 gives A096376, n >= 1.
Right border gives A000384, n >= 1.
(End)

C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 177.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Properties of Numbers, 1972.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
Index entries for sequences generated by sieves

Complement of A007607.
Cf. A007950, A007951, A007952, A048859, A004201.
Cf. A000384, A005917, A084849, A096376.

a007606 n = a007606_list !! (n-1)
a007606_list = takeSkip 1 [1..] where
   takeSkip k xs = take k xs ++ takeSkip (k + 2) (drop (2*k + 1) xs)
-- Reinhard Zumkeller, Feb 12 2011

Flatten[ Table[i, {j, 1, 17, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* Robert G. Wilson v, Mar 11 2004 *)
Join[{1},Flatten[With[{nn=20},Range[#[[1]],Total[#]]&/@Take[Thread[ {Accumulate[ Range[nn]]+1,Range[nn]}],{2,-1,2}]]]] (* Harvey P. Dale, Jun 23 2013 *)
With[{nn=20},Take[TakeList[Range[(nn(nn+1))/2],Range[nn]],{1,nn,2}]]//Flatten (* Harvey P. Dale, Feb 10 2023 *)

for(n=1,66,m=sqrtint(n-1);print1(n+m*(m+1),","))

a(n) = n + m*(m+1) where m = floor(sqrt(n-1)). - Klaus Brockhaus, Mar 26 2004

a(n+1) = a(n) + if n=k^2 then 2*k+1 else 1; a(1) = 1. - Reinhard Zumkeller, May 13 2009

A204164 Symmetric matrix based on f(i,j) = floor((i+j)/2), by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Clark Kimberling, Jan 12 2012

A204164 represents the matrix M given by f(i,j) = floor((i+j)/2) for i >= 1 and j >= 1.  See A204165 for characteristic polynomials of principal submatrices of M, with interlacing zeros.  See A204016 for a guide to other choices of M.
k appears 4k-1 times, k > 0. - Boris Putievskiy, Jun 12 2024
Number of numbers of the form 2k^2+k+1 <= n, for k = 0,1,2,... - Wesley Ivan Hurt, Jun 19 2024

			Northwest corner:
  1 1 2 2 3 3 4 4
  1 2 2 3 3 4 4 5
  2 2 3 3 4 4 5 5
  2 3 3 4 4 5 5 6
  3 3 4 4 5 5 6 6

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.

Cf. A057211, A204165, A204016, A202453, A370655.

f[i_, j_] := Floor[(i + j)/2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i], {n, 1, 15}, {i, 1, n}]]  (* this sequence *)
  (* or *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                 (* A204165 *)
TableForm[Table[c[n], {n, 1, 10}]]
  (* or *)
a[n_] = Ceiling[(Sqrt[8*n + 1] - 1)/4];
Nmax = 21; Table[a[n], {n, 1, Nmax}] (* Boris Putievskiy, Jun 12 2024 *)

from math import isqrt
def A204164(n): return (m:=isqrt(n>>1))+(n>m*((m<<1)+1)) # Chai Wah Wu, Nov 14 2024

a(n) = ceiling((sqrt(8*n+1)-1)/4). - Boris Putievskiy, Jun 12 2024
a(n) = Sum_{k=1..n} [c(k) = c(k-1)+1], where c(n) = floor(sqrt(2n)+1/2) mod 2 = A057211(n) and [] is the Iverson bracket. - Wesley Ivan Hurt, Jun 23 2024
a(n) = m+1 if n>m(2m+1) and a(n) = m otherwise where m = floor(sqrt(n/2)). - Chai Wah Wu, Nov 14 2024

A007607 Skip 1, take 2, skip 3, etc.

Original entry on oeis.org

2, 3, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21, 29, 30, 31, 32, 33, 34, 35, 36, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Numbers k with the property that the smallest Dyck path of the symmetric representation of sigma(k) has a central peak. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

Union of A317303 and A014105. - Omar E. Pol, Aug 29 2018

			From _Omar E. Pol_, Aug 29 2018: (Start)
Written as an irregular triangle in which the row lengths are the nonzero even numbers the sequence begins:
    2,   3;
    7,   8,   9,  10;
   16,  17,  18,  19,  20,  21;
   29,  30,  31,  32,  33,  34,  35,  36;
   46,  47,  48,  49,  50,  51,  52,  53,  54,  55;
   67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77,  78;
   92,  93,  94,  95,  96,  97,  98,  99, 100, 101, 102, 103, 104, 105;
  121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136;
...
Row sums give the nonzero terms of A317297.
Column 1 gives A130883, n >= 1.
Right border gives A014105, n >= 1.
(End)

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 177.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A063656, A004202, A063657, A007606, A064801, A004202.
Complement of A007606.
Cf. A014105, A130883, A317297.
Similar to A360418.

a007607 n = a007607_list !! (n-1)
a007607_list = skipTake 1 [1..] where
   skipTake k xs = take (k + 1) (drop k xs)
                   ++ skipTake (k + 2) (drop (2*k + 1) xs)
-- Reinhard Zumkeller, Feb 12 2011

a007607_list' = f $ tail $ scanl (+) 0 [1..] where
   f (t:t':t'':ts) = [t+1..t'] ++ f (t'':ts)
-- Reinhard Zumkeller, Feb 12 2011

Flatten[ Table[i, {j, 2, 16, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* Robert G. Wilson v, Mar 11 2004 *)
With[{t=20},Flatten[Take[TakeList[Range[(t(t+1))/2],Range[t]],{2,-1,2}]]] (* Harvey P. Dale, Sep 26 2021 *)

for(m=0,10,for(n=2*m^2+3*m+2,2*m^2+5*m+3,print1(n", "))) \\ Charles R Greathouse IV, Feb 12 2011

G.f.: 1/(1-x) * (1/(1-x) + x*Sum_{k>=1} (2k+1)*x^(k*(k+1))). - Ralf Stephan, Mar 03 2004
a(A000290(n)) = A001105(n). - Reinhard Zumkeller, Feb 12 2011
A057211(a(n)) = 0. - Reinhard Zumkeller, Dec 30 2011
a(n) = floor(sqrt(n) + 1/2)^2 + n = A053187(n) + n. - Ridouane Oudra, May 04 2019

A057212 n-th run has length n.

Original entry on oeis.org

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

Offset: 1

Ben Tyner (tyner(AT)phys.ufl.edu), Sep 27 2000

T(n,k) = 1 - n mod 2, 1 <= k <= n. [Reinhard Zumkeller, Mar 18 2011]

{a(n)} interpreted as a string over {0,1} is one of exactly two fixed-points of the function defined by f(0^n 1 s) = 1^(n-1) f(1 s) and f(1^n 0 s) = 0^(n-1) f(0 s). The other fixed point is obtained by swapping all 0s and 1s. - Curtis Bechtel, Jun 27 2025

K. H. Rosen, Discrete Mathematics and its Applications, 1999, fourth edition, page 79, exercise 10 (g).

Cf. A057211.

As a simple triangular or square array virtually the only sequences which appear are A000004, A000012 and A000035. Cf. A060510.

a057212 n = a057212_list !! (n-1)
a057212_list = concat $ zipWith ($) (map replicate [1..]) a000035_list
-- Reinhard Zumkeller, Mar 18 2011

A002024 := n->round(sqrt(2*n)):A057212 := n->(1+(-1)^A002024(n))/2;
# alternative Maple program:
T:= n-> [irem(1+n, 2)$n][]:
seq(T(n), n=1..14);  # Alois P. Heinz, Oct 06 2021

Table[If[OddQ[n], 0, 1], {n, 1, 14}, {n}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

from math import isqrt
def A057212(n): return int(not isqrt(n<<3)+1&2) # Chai Wah Wu, Jun 19 2024

a(n)=A003056(n) mod 2 so as a square array T(n, k)=n+k mod 2 - Henry Bottomley, Mar 22 2001
a(n) = (1+(-1)^A002024(n))/2, where A002024(n)=round(sqrt(2*n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003
a(n)=A163334(n) mod 2 = A163336(n) mod 2 = A163357(n) mod 2 = A163359(n) mod 2, i.e. the array gives the parity of elements at the successive antidiagonals (alternating between 0 and 1) of square arrays constructed from ANY Hilbert curve starting from zero located at the top left corner of a square grid (and using only N,E,S,W steps of length one). - Antti Karttunen, Oct 22 2012
a(n) = 1 - A057211(n). - Alois P. Heinz, Oct 06 2021

A138149 n-th run has length n-th prime, with values 0 and 1 only, starting with 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 29 2008

			.n ..... Run ....................... Length
.1 ..... 1,1 ....................... 2
.2 ..... 0,0,0 ..................... 3
.3 ..... 1,1,1,1,1 ................. 5
.4 ..... 0,0,0,0,0,0,0 ............. 7
.5 ..... 1,1,1,1,1,1,1,1,1,1,1 ..... 11

Cf. A000040, A057211.

Flatten[Table[{Table[1,{Prime[n]}],Table[0,{Prime[n+1]}]},{n,1,11,2}]] (* Harvey P. Dale, Jan 25 2013 *)

A138709 n-th run has length n-th positive Fibonacci numbers, with digits 0 and 1 only, starting with 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 29 2008

			n ..... Run ................. Length
.1 ..... 1 ................... 1
.2 ..... 0 ................... 1
.3 ..... 1,1 ................. 2
.4 ..... 0,0,0 ............... 3
.5 ..... 1,1,1,1,1 ........... 5
.6 ..... 0,0,0,0,0,0,0,0 ..... 8

Cf. A000045, A057211, A138149.

A138711 n-th run has length n-th positive triangular number, with digits 0 and 1 only, starting with 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 29 2008

			.n ..... Run ............................... Length
.1 ..... 1 ................................. 1
.2 ..... 0,0,0 ............................. 3
.3 ..... 1,1,1,1,1,1........................ 6
.4 ..... 0,0,0,0,0,0,0,0,0,0 ............... 10
.5 ..... 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ..... 15

Cf. A000217, A057211, A138149, A138709.

Flatten[Table[Join[PadRight[{},(n(n+1))/2,1],PadRight[{},((n+1)(n+2))/2, 0]],{n,1,7,2}]] (* Harvey P. Dale, Dec 11 2011 *)

cp's OEIS Frontend

A001614 Connell sequence: 1 odd, 2 even, 3 odd, ...

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A193356 If n is even then 0, otherwise n.

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A060734 Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are read by antidiagonals downwards.

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A007606 Take 1, skip 2, take 3, etc.

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A204164 Symmetric matrix based on f(i,j) = floor((i+j)/2), by antidiagonals.

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A007607 Skip 1, take 2, skip 3, etc.

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