A079979 -id:A079979 - cp's OEIS frontend

A000007 The characteristic function of {0}: a(n) = 0^n.

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

Offset: 0

N. J. A. Sloane

Changing the offset to 1 gives the arithmetical function a(1) = 1, a(n) = 0 for n > 1, the identity function for Dirichlet multiplication (see Apostol). - N. J. A. Sloane
Changing the offset to 1 makes this the decimal expansion of 1. - N. J. A. Sloane, Nov 13 2014
Hankel transform (see A001906 for definition) of A000007 (powers of 0), A000012 (powers of 1), A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001018 (powers of 8), A001019 (powers of 9), A011557 (powers of 10), A001020 (powers of 11), etc. - Philippe Deléham, Jul 07 2005
This is the identity sequence with respect to convolution. - David W. Wilson, Oct 30 2006
a(A000004(n)) = 1; a(A000027(n)) = 0. - Reinhard Zumkeller, Oct 12 2008
The alternating sum of the n-th row of Pascal's triangle gives the characteristic function of 0, a(n) = 0^n. - Daniel Forgues, May 25 2010
The number of maximal self-avoiding walks from the NW to SW corners of a 1 X n grid. - Sean A. Irvine, Nov 19 2010
Historically there has been some disagreement as to whether 0^0 = 1. Graphing x^0 seems to support that conclusion, but graphing 0^x instead suggests that 0^0 = 0. Euler and Knuth have argued in favor of 0^0 = 1. For some calculators, 0^0 triggers an error, while in Mathematica, 0^0 is Indeterminate. - Alonso del Arte, Nov 15 2011
Another consequence of changing the offset to 1 is that then this sequence can be described as the sum of Moebius mu(d) for the divisors d of n. - Alonso del Arte, Nov 28 2011
With the convention 0^0 = 1, 0^n = 0 for n > 0, the sequence a(n) = 0^|n-k|, which equals 1 when n = k and is 0 for n >= 0, has g.f. x^k. A000007 is the case k = 0. - George F. Johnson, Mar 08 2013
A fixed point of the run length transform. - Chai Wah Wu, Oct 21 2016

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

David Wasserman, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Dr. Math, 0^0 (zero to the zero power)
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Donald E. Knuth, Two notes on notation, arXiv:math/9205211 [math.HO], 1992. See page 6 on 0^0.
Robert Price, Comments on A000007, Jan 27 2016
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for "core" sequences
Index entries for characteristic functions
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (1).

Characteristic function of {g}: this sequence (g = 0), A063524 (g = 1), A185012 (g = 2), A185013 (g = 3), A185014 (g = 4), A185015 (g = 5), A185016 (g = 6), A185017 (g = 7). - Jason Kimberley, Oct 14 2011
Characteristic function of multiples of g: this sequence (g = 0), A000012 (g = 1), A059841 (g = 2), A079978 (g = 3), A121262 (g = 4), A079998 (g = 5), A079979 (g = 6), A082784 (g = 7). - Jason Kimberley, Oct 14 2011
Cf. A074909, A027641, A057427.

a000007 = (0 ^)
a000007_list = 1 : repeat 0
-- Reinhard Zumkeller, May 07 2012, Mar 27 2012

[1] cat [0:n in [1..100]]; // Sergei Haller, Dec 21 2006

A000007 := proc(n) if n = 0 then 1 else 0 fi end: seq(A000007(n), n=0..20);
spec := [A, {A=Z} ]: seq(combstruct[count](spec, size=n+1), n=0..20);

Table[If[n == 0, 1, 0], {n, 0, 99}]
Table[Boole[n == 0], {n, 0, 99}] (* Michael Somos, Aug 25 2012 *)
Join[{1},LinearRecurrence[{1},{0},102]] (* Ray Chandler, Jul 30 2015 *)
PadRight[{1},120,0] (* Harvey P. Dale, Jul 18 2024 *)

PARI
```
{a(n) = !n};
```

def A000007(n): return int(n==0) # Chai Wah Wu, Feb 04 2022

Multiplicative with a(p^e) = 0. - David W. Wilson, Sep 01 2001
a(n) = floor(1/(n + 1)). - Franz Vrabec, Aug 24 2005
As a function of Bernoulli numbers (cf. A027641: (1, -1/2, 1/6, 0, -1/30, ...)), triangle A074909 (the beheaded Pascal's triangle) * B_n as a vector = [1, 0, 0, 0, 0, ...]. - Gary W. Adamson, Mar 05 2012
a(n) = Sum_{k = 0..n} exp(2*Pi*i*k/(n+1)) is the sum of the (n+1)th roots of unity. - Franz Vrabec, Nov 09 2012
a(n) = (1-(-1)^(2^n))/2. - Luce ETIENNE, May 05 2015
a(n) = 1 - A057427(n). - Alois P. Heinz, Jan 20 2016
From Ilya Gutkovskiy, Sep 02 2016: (Start)
Binomial transform of A033999.
Inverse binomial transform of A000012. (End)

A059841 Period 2: Repeat [1,0]. a(n) = 1 - (n mod 2); Characteristic function of even numbers.

Original entry on oeis.org

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

Offset: 0

Alford Arnold, Feb 25 2001

When viewed as a triangular array, the row sum values are 0 1 1 1 2 3 3 3 4 5 5 5 6 ... (A004525).
This is the r=0 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.
Successive binomial transforms of this sequence: A011782, A007051, A007582, A081186, A081187, A081188, A081189, A081190, A060531, A081192.
Characteristic function of even numbers: a(A005843(n))=1, a(A005408(n))=0. - Reinhard Zumkeller, Sep 29 2008
This sequence is the Euler transformation of A185012. - Jason Kimberley, Oct 14 2011
a(n) is the parity of n+1. - Omar E. Pol, Jan 17 2012
Read as partial sequences, we get to A000975. - Jon Perry, Nov 11 2014
Elementary Cellular Automata rule 77 produces this sequence.  See Wolfram, Weisstein and Index links below. - Robert Price, Jan 30 2016
Column k = 1 of A051159. - John Keith, Jun 28 2021
When read as a constant: decimal expansion of 10/99, binary expansion of 2/3. - Jason Bard, Aug 25 2025

			Triangle begins:
  1;
  0, 1;
  0, 1, 0;
  1, 0, 1, 0;
  1, 0, 1, 0, 1;
  0, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 0, 1, 0;
  1, 0, 1, 0, 1, 0, 1, 0;
  1, 0, 1, 0, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Atsuto Seko, Atsushi Togo, and Isao Tanaka, Group-theoretical high-order rotational invariants for structural representations: Application to linearized machine learning interatomic potential, arXiv:1901.02118 [physics.comp-ph], 2019.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (0,1).
Index entries for characteristic functions

One's complement of A000035 (essentially the same, but shifted once).
Cf. A033999 (first differences), A008619 (partial sums), A004525, A011782 (binomial transf.), A000975.
Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), this sequence (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), A079979 (g=6), A082784 (g=7).

a059841 n = (1 -) . (`mod` 2)
a059841_list = cycle [1,0]
-- Reinhard Zumkeller, May 05 2012, Dec 30 2011

[0^(n mod 2): n in  [0..100]]; // Vincenzo Librandi, Nov 09 2014

seq(1-modp(n,2), n=0..150); # Muniru A Asiru, Apr 05 2018

CoefficientList[Series[1/(1 - x^2), {x, 0, 104}], x] (* or *)
Array[1/2 + (-1)^#/2 &, 105, 0] (* Michael De Vlieger, Feb 19 2019 *)
Table[QBinomial[n, 1, -1], {n, 1, 74}] (* John Keith, Jun 28 2021 *)
PadRight[{},120,{1,0}] (* Harvey P. Dale, Mar 06 2023 *)

PARI
```
a(n)=(n+1)%2; \\ or 1-n%2 as in NAME.
```

A059841(n)=!bittest(n,0) \\ M. F. Hasler, Jan 13 2012

def A059841(n): return 1 - (n & 1) # Chai Wah Wu, May 25 2022

a(n) = 1 - A000035(n). - M. F. Hasler, Jan 13 2012
From Paul Barry, Mar 11 2003: (Start)
G.f.: 1/(1-x^2).
E.g.f.: cosh(x).
a(n) = (n+1) mod 2.
a(n) = 1/2 + (-1)^n/2. (End)
Additive with a(p^e) = 1 if p = 2, 0 otherwise.
a(n) = Sum_{k=0..n} (-1)^k*A038137(n, k). - Philippe Deléham, Nov 30 2006
a(n) = Sum_{k=1..n} (-1)^(n-k) for n > 0. - William A. Tedeschi, Aug 05 2011
E.g.f.: cosh(x) = 1 + x^2/(Q(0) - x^2); Q(k) = 8k + 2 + x^2/(1 + (2k + 1)*(2k + 2)/Q(k + 1)); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
E.g.f.: cosh(x) = 1/2*Q(0); Q(k) = 1 + 1/(1 - x^2/(x^2 + (2k + 1)*(2k + 2)/Q(k + 1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
E.g.f.: cosh(x) = E(0)/(1-x) where E(k) = 1 - x/(1 - x/(x - (2*k+1)*(2*k+2)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
For the general case: the characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = A000035(n+1) = A008619(n) - A110654(n). - Wesley Ivan Hurt, Jul 20 2013

Better definition from M. F. Hasler, Jan 13 2012

Reinhard Zumkeller's Sep 29 2008 description added as a secondary name by Antti Karttunen, May 03 2022

A079978 Characteristic function of multiples of three.

Original entry on oeis.org

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

Offset: 0

Vladimir Baltic, Feb 17 2003

Period 3: repeat [1, 0, 0].
a(n)=1 if n=3k, a(n)=0 otherwise.
Decimal expansion of 1/999.
Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=2, I={0,1}.
a(n) is also the number of partitions of n with every part being three (a(0)=1 because the empty partition has no parts). Hence a(n) is also the number of 2-regular graphs on n vertices with each component having girth 3. - Jason Kimberley, Oct 02 2011
Euler transformation of A185013. - Jason Kimberley, Oct 02 2011
If b(0)=0 and for n > 0, b(n)=a(n), then starting at n=0, b(n) is the number of incongruent equilateral triangles formed from the vertices of a regular n-gon. The number of incongruent isosceles triangles (strictly two equal sides) is A174257(n) and the number of incongruent scalene triangles is A069905(n-3) for n > 2, otherwise 0. The total number of incongruent triangles is A069905(n). - Frank M Jackson, Nov 19 2022

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135.
N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Proceedings of Applications of Computer Algebra ACA, 2013.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.
Essentially the same as A022003.
Partial sums are given by A002264(n+3).
Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), this sequence (g=3), A121262 (g=4), A079998 (g=5), A079979 (g=6), A082784 (g=7). - Jason Kimberley, Oct 14 2011
Cf. A007908, A011655 (bit flipped).

a079978 = fromEnum . (== 0) . (`mod` 3)
a079978_list = cycle [1,0,0]
-- Reinhard Zumkeller, Aug 28 2012, Nov 26 2011

&cat[[1,0,0]^^30]; // Vincenzo Librandi, Dec 26 2015

seq(op([1, 0, 0]), n=0..50); # Wesley Ivan Hurt, Jun 30 2016

Table[Boole[IntegerQ[n/3]], {n, 0, 127}] (* Michael De Vlieger, Jan 03 2015, after Alonso del Arte at A121262 *)

a(n)=!(n%3) \\ Jaume Oliver Lafont, Mar 01 2009

a(n) = a(n-3) for n > 2.
G.f.: 1/(1-x^3) = 1/( (1-x)*(1+x+x^2)).
a(n) = (1 + e^(i*Pi*A002487(n)))/2, i=sqrt(-1). - Paul Barry, Jan 14 2005
Additive with a(p^e) = 1 if p = 3, 0 otherwise.
a(n) = ((n+1) mod 3) mod 2. Also: a(n) = (1/2)*(1 + (-1)^(n + floor((n+1)/3))). - Hieronymus Fischer, May 29 2007
a(n) = 1 - A011655(n). - Reinhard Zumkeller, Nov 30 2009
a(n) = (1 + (-1)^(2*n/3) + (-1)^(-2*n/3))/3. - Jaume Oliver Lafont, May 13 2010
For the general case: the characteristic function of numbers that are multiples of m is a(n) = floor(n/m) - floor((n-1)/m), m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = floor( ((n-1) mod 3)/2 ). - Wesley Ivan Hurt, Jun 29 2013
a(n) = 2^(n mod 3) mod 2. - Olivier Gérard, Jul 04 2013
a(n) = (w^(2*n) + w^n + 1)/3, w = (-1 + i*sqrt(3))/2 (w is a primitive 3rd root of unity). - Bogart B. Strauss, Jul 20 2013
E.g.f.: (exp(x) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/3. - Geoffrey Critzer, Nov 03 2014
a(n) = (sin(Pi*(n+1)/3)^2)*(2/3) + sin(Pi*(n+1)*2/3)/sqrt(3). - Mikael Aaltonen, Jan 03 2015
a(n) = (2*n^2 + 1) mod 3. The characteristic function of numbers that are multiples of 2k+1 is (2*k*n^(2*k) + 1) mod (2k+1). Example: A058331(n) mod 3 for k=1, A211412(n) mod 5 for k=2, ... - Eric Desbiaux, Dec 25 2015
a(n) = floor(2*(n-1)/3) - 2*floor((n-1)/3). - Wesley Ivan Hurt, Jul 25 2016
a(n) == A007908(n+1) (mod 3), n >= 0. See A011655 (bit flipped). - Wolfdieter Lang, Jun 12 2017
a(n) = 1/3 + (2/3)*cos((2/3)*n*Pi). - Ridouane Oudra, Jan 22 2021
a(n) = A000217(n+1) mod 3. - Christopher Adams, Jan 05 2025

Name simplified by Ralf Stephan, Nov 22 2010

Name changed by Jason Kimberley, Oct 14 2011

A121262 The characteristic function of the multiples of four.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2006, Aug 30 2007

Period 4: repeat [1, 0, 0, 0].
a(n) is also the number of partitions of n where each part is four (Since the empty partition has no parts, a(0) = 1). Hence a(n) is also the number of 2-regular graphs on n vertices such that each component has girth exactly four. - Jason Kimberley, Oct 01 2011
This sequence is the Euler transformation of A185014. - Jason Kimberley, Oct 01 2011
Number of permutations satisfying -k <= p(i) - i <= r and p(i)-i not in I, i = 1..n, with k = 1, r = 3, I = {0, 1, 2}. - Vladimir Baltic, Mar 07 2012

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 82.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135
Steve Chow, 0,0,0,1,0,0,0,1 (deriving an explicit formula for the sequence) :YouTube Video, 2017.
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

A011765 is another version of the same sequence.
Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), this sequence (g=4), A079998 (g=5), A079979 (g=6), A082784 (g=7). - Jason Kimberley, Oct 14 2011
Cf. A010873, A166486, A185014.

a121262 = (0 ^) . flip mod 4  -- Reinhard Zumkeller, Mar 04 2015
a121262_list = cycle [1,0,0,0]  -- Reinhard Zumkeller, Jan 06 2012

&cat [[1, 0, 0, 0]^^30]; // Wesley Ivan Hurt, Jul 07 2016

seq(op([1, 0, 0, 0]), n=0..50); # Wesley Ivan Hurt, Jul 07 2016

Table[Boole[IntegerQ[n/4]], {n,0,127}] (* Alonso del Arte, Jul 14 2013 *)

a(n)=!(n%4) \\ Charles R Greathouse IV, Oct 25 2012

a(n) = (1/4)*(2*cos(n*Pi/2) + 1 + (-1)^n).
Additive with a(p^e) = 1 if p = 2 and e > 1, 0 otherwise.
Sequence shifted right by 2 is additive with a(p^e) = 1 if p = 2 and e = 1, 0 otherwise.
a(n) = 1 - (C(n + 1, n + (-1)^(n+1)) mod 2).
a(n) = 0^(n mod 4). - Reinhard Zumkeller, Sep 30 2008
a(n) = !(n%4). - Jaume Oliver Lafont, Mar 01 2009
a(n) = (1/4)*(1 + I^n + (-1)^n + (-I)^n). - Paolo P. Lava, May 04 2010
a(n) = ((n-1)^k mod 4 - (n-1)^(k-1) mod 4)/2, k > 2. - Gary Detlefs, Feb 21 2011
a(n) = floor(1/2*cos(n*Pi/2) + 1/2). - Gary Detlefs, May 16 2011
G.f.: 1/(1 - x^4); a(n) = (1 + (-1)^n)*(1 + i^((n-1)*n))/4, where i = sqrt(-1). - Bruno Berselli, Sep 28 2011
a(n) = floor(((n+3) mod 4)/3). - Gary Detlefs, Dec 29 2011
a(n) = floor(n/4) - floor((n-1)/4). - Tani Akinari, Oct 25 2012
a(n) = ceiling( (1/2)*cos(Pi*n/2) ). - Wesley Ivan Hurt, May 31 2013
a(n) = ((1+(-1)^(n/2))*(1+(-1)^n))/4. - Bogart B. Strauss, Jul 14 2013
a(n) = C(n-1,3) mod 2. - Wesley Ivan Hurt, Oct 07 2014
a(n) = (((n+1) mod 4) mod 3) mod 2. - Ctibor O. Zizka, Dec 11 2014
a(n) = (sin(Pi*(n+1)/2)^2)/2 + sin(Pi*(n+1)/2)/2. - Mikael Aaltonen, Jan 02 2015
E.g.f.: (cos(x) + cosh(x))/2. - Vaclav Kotesovec, Feb 15 2015
a(n) = a(n-4) for n>3. - Wesley Ivan Hurt, Jul 07 2016
a(n) = (1-sqrt(2)*cos(n*Pi/2-3*Pi/4))/2 * cos(n*Pi/2). - (found by Steve Chow) Iain Fox, Nov 16 2017
a(n) = 1-A166486(n). - Antti Karttunen, Jul 29 2018
a(n) = (1-(-1)^A142150(n+1))/2. - Adriano Caroli, Sep 28 2019

More terms from Antti Karttunen, Jul 29 2018

A010875 a(n) = n mod 6.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0

Offset: 0

N. J. A. Sloane

Period 6: repeat [0, 1, 2, 3, 4, 5].
The rightmost digit in the base-6 representation of n. - Hieronymus Fischer, Jun 11 2007
[a(n) * a(m)] mod 6 == a(n*m mod 6) == a(n*m). - Jon Perry, Nov 11 2014
If n > 3 and (a(n) is in {0,2,3,4}), then n is not prime. - Jean-Marc Rebert, Jul 22 2015, corrected by M. F. Hasler, Jul 24 2015

Antti Karttunen, Table of n, a(n) for n = 0..65538
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Partial sums: A130484. Other related sequences A130481, A130482, A130483, A130485.

Cf. also A079979, A097325, A122841.

[n mod 6: n in [0..100]]; // Wesley Ivan Hurt, Jul 06 2014

A010875:=n->n mod 6; seq(A010875(n), n=0..100); # Wesley Ivan Hurt, Jul 06 2014

Mod[Range[0, 100], 6] (* Wesley Ivan Hurt, Jul 06 2014 *)

a(n)=n%6 \\ Charles R Greathouse IV, Dec 05 2011

[power_mod(n,3,6 )for n in range(0, 81)] # Zerinvary Lajos, Oct 29 2009

(define (A010875 n) (modulo n 6)) ;; Antti Karttunen, Dec 22 2017

Complex representation: a(n) = (1/6) * (1 - r^n) * Sum_{k = 1..6} k * Product_{1 <= m < 6, m <> k} (1-r^(n-m)), where r = exp((Pi/3)*i) = (1 + sqrt(3)*i)/2 and i = sqrt(-1).
Trigonometric representation: a(n) = (16/3)^2 * (sin(n*Pi/6))^2 * Sum_{k = 1..6} k * Product_{1 <= m < 6, m<>k} (sin((n-m)*Pi/6))^2.
G.f.: g(x) = (Sum_{k = 1..6} k*x^k)/(1-x^6).
Also: g(x) = x*(5*x^6 - 6*x^5 + 1)/((1 - x^6)*(1 - x)^2). - Hieronymus Fischer, May 31 2007
a(n) = (n mod 2) + 2(floor(n/2) mod 3) = A000035(n) +2*A010872(A004526(n));
a(n) = (n mod 3) + 3(floor(n/3) mod 2) = A010872(n) +3*A000035(A002264(n)). - Hieronymus Fischer, Jun 11 2007
a(n) = 2.5 - 0.5*(-1)^n - cos(Pi*n/3) - 3^0.5*sin(Pi*n/3) -cos(2*Pi*n/3) - 3^0.5/3*sin(2*Pi*n/3). - Richard Choulet, Dec 11 2008
a(n) = n^3 mod 6. - Zerinvary Lajos, Oct 29 2009
a(n) = floor(12345/999999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = floor(373/9331*6^(n+1)) mod 6. - Hieronymus Fischer, Jan 04 2013
a(n) = 5/2 - (-1)^n/2 - 2*0^((-1)^(n/6 - 1/12 + (-1)^n/12) - (-1)^(n/2 - 1/4 +(-1)^n/4)) + 2*0^((-1)^(n/6 + 1/4 + (-1)^n/12) + (-1)^(n/2 - 1/4 + (-1)^n/4)). - Wesley Ivan Hurt, Jun 23 2015
E.g.f.: -sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2) - 2*cosh(x/2)*cos(sqrt(3)*x/2). - Robert Israel, Jul 22 2015

Formulas 1 to 6 re-edited for better readability by Hieronymus Fischer, Dec 05 2011

More terms from Antti Karttunen, Dec 22 2017

A079998 The characteristic function of the multiples of five.

Original entry on oeis.org

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

Offset: 0

Vladimir Baltic, Feb 10 2003

Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i = 1..n, with k = 2, r = 3, I = {-1, 0, 1, 2}.
a(n) = 1 if n = 5k, a(n) = 0 otherwise. Also, number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i = 1..n, with k = 1, r = 4, I = {0, 1, 2, 3}.
a(n) is also the number of partitions of n with each part being five (a(0) = 1 because the empty partition has no parts to test equality with five). Hence a(n) is also the number of 2-regular graphs on n vertices with each component having girth exactly five. - Jason Kimberley, Oct 02 2011
This sequence is the Euler transformation of A185015. - Jason Kimberley, Oct 02 2011

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Antti Karttunen, Table of n, a(n) for n = 0..16385
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135.
Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 10.
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A011558, A008587, A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Characteristic function of multiples of g: A000007 (g = 0), A000012 (g = 1), A059841 (g = 2), A079978 (g = 3), A121262 (g = 4), this sequence (g = 5), A079979 (g = 6), A082784 (g = 7). - Jason Kimberley, Oct 14 2011

[Binomial(n-1,4) mod 5 : n in [0..100]]; // Wesley Ivan Hurt, Oct 06 2014

A079998:=n->binomial(n-1,4) mod 5: seq(A079998(n), n=0..100); # Wesley Ivan Hurt, Oct 06 2014

Table[Mod[Binomial[n - 1, 4], 5], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 06 2014 *)
Table[Boole[Divisible[n, 5]], {n, 0, 99}] (* Alonso del Arte, Nov 29 2014 *)
PadRight[{},120,{1,0,0,0,0}] (* Harvey P. Dale, Jul 11 2023 *)

a(n)=!(n%5) \\ Charles R Greathouse IV, Mar 07 2012

(define (A079998 n) (if (zero? (modulo n 5)) 1 0)) ;; Antti Karttunen, Dec 21 2017

Recurrence: a(n) = a(n-5). G.f.: -1/(x^5 - 1).
a(n) = 1 - A011558(n); a(A008587(n)) = 1; a(A047201(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
a(n) = floor(1/2*cos(2*n*Pi/5) + 1/2). - Gary Detlefs, May 16 2011
a(n) = floor(n/5) - floor((n-1)/5). - Tani Akinari, Oct 21 2012
a(n) = binomial(n - 1, 4) mod 5. - Wesley Ivan Hurt, Oct 06 2014

More terms from Antti Karttunen, Dec 21 2017

A088911 Period 6: repeat [1, 1, 1, 0, 0, 0].

Original entry on oeis.org

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

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Oct 22 2003

For periodic sequences having a period of 2*k and composed of k ones followed by k zeros we have a(n) = floor(((n+k) mod 2*k)/k). Sequences of this form are A000035(n+1) (k=1), A133872(n) (k=2), this sequence (k=3), A131078(n) (k=4), and A112713(n-1) (k=5). - Gary Detlefs, May 17 2011

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A000035, A010701, A079979, A133872, A131078, A112713.

&cat [[1, 1, 1, 0, 0, 0]^^30]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([1, 1, 1, 0, 0, 0]), n=0..40); # Wesley Ivan Hurt, Jul 05 2016

CoefficientList[Series[(1 + x + x^2)/(1 - x^6), {x, 0, 50}], x]
Flatten[Table[{1,1,1,0,0,0},{20}]] (* Harvey P. Dale, Jul 17 2011 *)

a(n)=n%6<3 \\ Jaume Oliver Lafont, Mar 17 2009

def A088911(n): return int(n % 6 < 3) # Chai Wah Wu, May 25 2022

G.f.: (1+x+x^2)/(1-x^6) = 1/((1-x)*(1+x)*(1-x+x^2)).
a(n) = a(n-6) for n>=6, a(0)=a(1)=a(2)=1, a(3)=a(4)=a(5)=0.
a(n) = ((-1)^floor((5*n + 2)/3) + 1)/2 = ( (-1)^floor(n/3) + 1 )/2. [Simplified by Bruno Berselli, Jul 09 2013]
a(n) = Sum_{k=0..floor(n/2)} U(n-2k, 1/2). - Paul Barry, Nov 15 2003
From Paul Barry, Mar 14 2004: (Start)
Partial sums of expansion of 1/(1+x^3), see A131531.
a(n) = 2*sin(Pi*n/3 + Pi/6)/3 + cos(Pi*n)/6 + 1/2. (End)
a(n) = floor(((n+3) mod 6)/3).
a(n) = floor((5*n-1)/3) mod 2. - Gary Detlefs, May 17 2011
a(n) = 1/2 + cos(Pi*n/3)/3 + sin(Pi*n/3)/sqrt(3) + (-1)^n/6. - R. J. Mathar, Oct 08 2011
a(n) = floor(((n+2)^2)/3) mod 2. - Wesley Ivan Hurt, Jun 29 2013
a(n) = A079979(n) + A079979(n-1) + A079979(n-2). - R. J. Mathar, Jul 10 2015
a(n) = a(n-1) - a(n-3) + a(n-4) for n > 3. - Wesley Ivan Hurt, Jul 05 2016
a(n) = 2*floor(n/6) - floor(n/3) + 1. - Ridouane Oudra, Dec 14 2021
E.g.f.: (2*cosh(x) + exp(x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + sinh(x))/3. - Stefano Spezia, Aug 04 2025

A082784 Characteristic function of multiples of 7.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, May 22 2003

This sequence is the Euler transformation of A185017. - Jason Kimberley, Oct 14 2011

			a(14) = a(2*7) = 1; a(41) = a(5*7+6) = 0.

Index entries for characteristic functions.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1).

Cf. A008589, A076309.
Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), A079979 (g=6), this sequence (g=7). - Jason Kimberley, Oct 14 2011
Cf. A000007, A047304, A109720, A185017.

a082784 = a000007 . (`mod` 7)
a082784_list = cycle [1,0,0,0,0,0,0]
-- Reinhard Zumkeller, Oct 27 2012

[Binomial(n-1,6) mod 7 : n in [0..100]]; // Wesley Ivan Hurt, Oct 07 2014

A082784:=n->0^(n mod 7): seq(A082784(n), n=0..100); # Wesley Ivan Hurt, Oct 07 2014

Table[Mod[Binomial[n - 1, 6], 7], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 07 2014 *)
Table[Boole[Divisible[n, 7]], {n, 0, 100}] (* Amiram Eldar, Oct 31 2023 *)

a(n)=!(n%7) \\ Charles R Greathouse IV, Dec 03 2012

a(n) = 0^(n mod 7).
a(0)=1, a(n)=0 for 1<=n<7, a(n+7)=a(n).
a(n) = 1 - (n^6 mod 7). - Paolo P. Lava, Oct 02 2006
a(n) = 1 - A109720(n); a(A008589(n)) = 1; a(A047304(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
a(n) = floor(n/7)-floor((n-1)/7). - Tani Akinari, Oct 26 2012
a(n) = C(n-1,6) mod 7. - Wesley Ivan Hurt, Oct 07 2014
From Wesley Ivan Hurt, Jul 11 2016: (Start)
G.f.: 1/(1-x^7).
a(n) = a(n-7) for n>6.
a(n) = (gcd(n,7) - 1)/6. (End)

Wrong formula and keyword mult removed by Amiram Eldar, Oct 31 2023

A097325 Period 6: repeat [0, 1, 1, 1, 1, 1].

Original entry on oeis.org

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

Offset: 0

Ralf Stephan, Aug 16 2004

a(n) is 0 if 6 divides n, 1 otherwise.

Antti Karttunen, Table of n, a(n) for n = 0..26244
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Characteristic sequence of A047253.
Binary complement of A079979.
Cf. A010875, A168185, A145568, A168184, A168182, A168181, A109720, A011558, A166486, A011655, A000035.

[Sign(n mod 6) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

seq(signum(k mod 6), k=0..100); # Wesley Ivan Hurt, Jun 29 2013

Table[Boole[Not[Divisible[n, 6]]], {n, 0, 89}] (* Alonso del Arte, Oct 21 2013 *)
PadRight[{}, 120, {0, 1, 1, 1, 1, 1}] (* Michael De Vlieger, Dec 22 2017 *)

PARI
```
a(n) = sign(n%6);
```

(define (A097325 n) (if (zero? (modulo n 6)) 0 1)) ;; Antti Karttunen, Dec 22 2017

G.f.: 1/(1-x) - 1/(1-x^6) = Sum_{k>=0} x^k - x^(6*k).
Recurrence: a(n+6) = a(n), a(0) = 0, a(i) = 1, 1 <= i <= 5.
a(n) = (1/4) * (3 - (-1)^n - (-1)^((n+1)/3) - (-1)^((2n+1)/3)).
From Reinhard Zumkeller, Nov 30 2009: (Start)
a(n) = 1 - A079979(n).
a(A047253(n)) = 1, a(A008588(n)) = 0.
A033438(n) = Sum_{k=0..n} a(k)*(n-k). (End)
Dirichlet g.f.: (1 - 1/6^s)*zeta(s). - R. J. Mathar, Feb 19 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m, n > 0. - Boris Putievskiy, May 08 2013
a(n) = sign(n mod 6). - Wesley Ivan Hurt, Jun 29 2013
a(n) = ceiling(5n/6) - floor(5n/6). - Wesley Ivan Hurt, Jun 20 2014

New name from Omar E. Pol, Oct 21 2013

A185016 Characteristic function of 6.

Original entry on oeis.org

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

Offset: 0

Jason Kimberley, Sep 30 2011

Number of connected 2-regular (simple) graphs with girth exactly 6.

The Euler transformation of this sequence is A079979.

Index entries for linear recurrences with constant coefficients, signature (1).

Characteristic function of g: A000007 (g=0), A063524 (g=1), A185012 (g=2), A185013 (g=3), A185014 (g=4), A185015 (g=5), this sequence (g=6), A185017 (g=7).

def A185016(n): return int(n==6) # Chai Wah Wu, Feb 04 2022

a(n) = A185116(n) - A185117(n).

cp's OEIS Frontend

A000007 The characteristic function of {0}: a(n) = 0^n.

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A059841 Period 2: Repeat [1,0]. a(n) = 1 - (n mod 2); Characteristic function of even numbers.

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A079978 Characteristic function of multiples of three.

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A121262 The characteristic function of the multiples of four.

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A010875 a(n) = n mod 6.

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