A079998 -id:A079998 - 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

A008587 Multiples of 5: a(n) = 5 * n.

Original entry on oeis.org

0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265, 270, 275

Offset: 0

N. J. A. Sloane

1/31 = 0.0322580645... = (1/2)^5 + (1/2)^10 + (1/2)^15 + ... - Gary W. Adamson, Mar 14 2009
Complement of A047201; A079998(a(n))=1; A011558(a(n))=0. - Reinhard Zumkeller, Nov 30 2009
The y-intercept of a line perpendicular to y=mx,where m is the slope a/b and in this case a=2 and b=1, is a^2 + b^2 or 5, the first value of the list given. The remaining value are multiples of the first number of the list. - Larry J Zimmermann, Aug 21 2010

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 85.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 317
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. index to numbers of the form n*(d*n+10-d)/2 in A140090.

Cf. A008706, A011558, A020714, A030308, A047201, A079998.

[5*n: n in [0..60]]; // Vincenzo Librandi, Jun 10 2013

Range[0, 500, 5] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
NestList[#+5&,0,60] (* Harvey P. Dale, Dec 18 2023 *)

makelist(5*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n)=5*n \\ Charles R Greathouse IV, Mar 22 2016

From R. J. Mathar, May 26 2008: (Start)
O.g.f.: 5x/(1-x)^2.
a(n) = A008706(n), n > 0. (End)
a(n) = Sum_{k>=0} A030308(n,k)*A020714(k). - Philippe Deléham, Oct 17 2011
E.g.f.: 5*x*exp(x). - Stefano Spezia, Aug 19 2024

A045572 Numbers that are odd but not divisible by 5.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143, 147, 149, 151, 153

Offset: 1

Jeff Burch, Dec 11 1999

Contains the repunits R_n, (A000042 or A002275): For any m in the sequence (divisible by neither 2 nor 5), Euler's theorem (i.e., m | 10^m - 1 = 9*R_n) guarantees that R_n is always some multiple of m (see A099679) and thus forms a subsequence. - Lekraj Beedassy, Oct 26 2004
Inverse formula: n = 4*floor(a(n)/10) + floor((a(n) mod 10)/3) + 1. - Carl R. White, Feb 06 2008
Numbers ending with 1, 3, 7 or 9. - Lekraj Beedassy, Apr 04 2009
Complement of A065502. - Reinhard Zumkeller, Nov 15 2009
Union of evenish and oddish numbers, cf. A045797, A045798. - Reinhard Zumkeller, Dec 10 2011
Numbers k such that k^(4*j) mod 10 = 1, for any j. - Gary Detlefs, Jan 03 2012
Numbers coprime to 10. - Charles R Greathouse IV, Sep 05 2013
This is also the sequence of numbers such that all their divisors are the sum of the proper divisors of some number (see A001065 (sum of proper divisors) and A078923 (possible values of sigma(n)-n)). This is due to the fact that in the set of untouchable numbers (A005114) there are only 2 prime numbers (2 and 5) and all other terms are even composite. - Michel Marcus, Jun 14 2014
Numbers n for which A001589(n) is divisible by 5. - Bruno Berselli, Jun 18 2014
For a(n) > 1, positive integers x such that the decimal representation of 1/x is purely periodic after the decimal point (1/x is a repeating decimal with no non-repeating portion). - Doug Bell, Aug 05 2015
The asymptotic density of this sequence is 2/5. - Amiram Eldar, Oct 18 2020

			a(18) = 10*floor(17/4) + 2*floor( (4*(17 mod 4) + 1)/3 ) + 1
      = 10*4 + 2*floor( (4*(1)+1)/3 ) + 1
      = 40 + 2*floor(5/3) + 1
      = 40 + 2*1 + 1
      = 43.
G.f. = x + 3*x^2 + 7*x^3 + 9*x^4 + 11*x^5 + 13*x^6 + 17*x^7 + 19*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Relative complement of A017329 in A005408.

Cf. A000035, A000042, A001065, A001589, A002275, A005114, A045797, A045798, A065502, A078923, A079998, A082768 (numbers that begin with 1, 3, 7 or 9), A085820, A099679.

a045572 n = a045572_list !! (n-1)
a045572_list = filter ((/= 0) . (`mod` 5)) a005408_list
-- Reinhard Zumkeller, Dec 10 2011

[ 2*n + 2*Floor((n-3)/4) + 1: n in [1..70] ]; // Vincenzo Librandi, Aug 01 2011

A045572:=n->2*n + 2*floor((n-3)/4) + 1; seq(A045572(n), n=1..50); # Wesley Ivan Hurt, Jun 14 2014

Flatten[Table[10n + {1, 3, 7, 9}, {n, 0, 19}]] (* Alonso del Arte, Jan 13 2012 *)
Select[2Range@78 -1, Mod[#, 5] > 0 &] (* Robert G. Wilson v, Apr 02 2017 *)
Map[(1/2*(5*# + Mod[3*# + 2, 4] - 4))&, Range[10^3]] (* Mikk Heidemaa, Nov 23 2017 *)

a(n)=10*(n>>2)+[-1,1,3,7][n%4+1] \\ Charles R Greathouse IV, Jul 31 2011

is(n)=gcd(n,10)==1 \\ Charles R Greathouse IV, Sep 05 2013

{a(n) = 2*n - 1 + (n+1) \ 4 * 2}; /* Michael Somos, Jun 15 2014 */

def A045572(n): return 2*(n+(n+1)//4) - 1 # Chai Wah Wu, Jan 08 2022

a(n) = 10*floor((n-1)/4) + 2*floor( (4*((n-1) mod 4) + 1)/3 ) + 1; a(n) = a(n-1) + 2 + 2*floor(((x+6) mod 10)/9). - Carl R. White, Feb 06 2008
a(n) = 2*n + 2*floor((n-3)/4) + 1. - Kenneth Hammond (weregoose(AT)gmail.com), Mar 07 2008
a(n) = -1 + 2*n + 2*floor((n+1)/4). - Kenneth Hammond (weregoose(AT)gmail.com), Mar 25 2008
From R. J. Mathar, Sep 22 2009: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3 + x^4)/((1+x) * (x^2+1) * (x-1)^2). (End)
A000035(a(n))*(1 - A079998(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2009
a(n) = (10*n + 2*(-1)^(n*(n+1)/2) - (-1)^n - 5)/4. - Bruno Berselli, Nov 06 2011
G.f.: x * (1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x) * (1 - x^4)). - Michael Somos, Jun 15 2014
a(1 - n) = -a(n) for all n in Z. - Michael Somos, Jun 15 2014
0 = (a(n) - 2*a(n+1) + a(n+2)) * (a(n) - 4*a(n+2) + 3*a(n+3)) for all n in Z. - Michael Somos, Jun 15 2014
From Mikk Heidemaa, Nov 22 2017: (Start)
a(n) = (1/2)*(5*n + ((3*n + 2) mod 4) - 4);
a(n) = (1/4)*((-1)^(n + 1) + 10*n + 2*cos((n*Pi)/2) - 2*sin((n*Pi)/2) - 5);
a(n) = (1/4)*((-1)^(1 + n) + (1 - i)*exp(-(1/2)*i*n*Pi) + (1 + i)*exp(i*n*Pi/2) + 10*n - 5) (for n > 0), where i is the imaginary unit. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(10-2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 12 2021
E.g.f.: (2 + cos(x) + (5*x - 3)*cosh(x) - sin(x) + (5*x - 2)*sinh(x))/2. - Stefano Spezia, Dec 07 2022

A127093 Triangle read by rows: T(n,k)=k if k is a divisor of n; otherwise, T(n,k)=0 (1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 2, 0, 4, 0, 0, 0, 8, 1, 0, 3, 0, 0, 0, 0, 0, 9, 1, 2, 0, 0, 5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, Jan 05 2007, Apr 04 2007

Sum of terms in row n = sigma(n) (sum of divisors of n).
Euler's derivation of A127093 in polynomial form is in his proof of the formula for Sigma(n): (let S=Sigma, then Euler proved that S(n) = S(n-1) + S(n-2) - S(n-5) - S(n-7) + S(n-12) + S(n-15) - S(n-22) - S(n-26), ...).
[Young, pp. 365-366], Euler begins, s = (1-x)*(1-x^2)*(1-x^3)*... = 1 - x - x^2 + x^5 + x^7 - x^12 ...; log s = log(1-x) + log(1-x^2) + log(1-x^3) ...; differentiating and then changing signs, Euler has t = x/(1-x) + 2x^2/(1-x^2) + 3x^3/(1-x^3) + 4x^4/(1-x^4) + 5x^5/(1-x^5) + ...
Finally, Euler expands each term of t into a geometric series, getting A127093 in polynomial form: t =
  x +  x^2 +  x^3 +  x^4 +  x^5 +  x^6 +  x^7 +  x^8 + ...
    + 2x^2        + 2x^4        + 2x^6        + 2x^8 + ...
           + 3x^3               + 3x^6               + ...
                  + 4x^4                      + 4x^8 + ...
                         + 5x^5                      + ...
                                + 6x^6               + ...
                                       + 7x^7        + ...
                                              + 8x^8 + ...
T(n,k) is the sum of all the k-th roots of unity each raised to the n-th power. - Geoffrey Critzer, Jan 02 2016
From Davis Smith, Mar 11 2019: (Start)
For n > 1, A020639(n) is the leftmost term, other than 0 or 1, in the n-th row of this array. As mentioned in the Formula section, the k-th column is period k: repeat [k, 0, 0, ..., 0], but this also means that it's the characteristic function of the multiples of k multiplied by k. T(n,1) = A000012(n), T(n,2) = 2*A059841(n), T(n,3) = 3*A079978(n), T(n,4) = 4*A121262(n), T(n,5) = 5*A079998(n), and so on.
The terms in the n-th row, other than 0, are the factors of n. If n > 1 and for every k, 1 <= k < n, T(n,k) = 0 or 1, then n is prime. (End)
From Gary W. Adamson, Aug 07 2019: (Start)
Row terms of the triangle can be used to calculate E(n) in A002654): (1, 1, 0, 1, 2, 0, 0, 1, 1, 2, ...), and A004018, the number of points in a square lattice on the circle of radius sqrt(n), A004018: (1, 4, 4, 0, 4, 8, 0, 0, 4, ...).
As to row terms in the triangle, let E(n) of even terms = 0,
  E(integers of the form 4*k - 1 = (-1), and E(integers of the form 4*k + 1 = 1.
Then E(n) is the sum of the E(n)'s of the factors of n in the triangle rows.  Example: E(10) = Sum: ((E(1) + E(2) + E(5) + E(10)) = ((1 + 0 + 1 + 0) = 2, matching A002654(10).
To get A004018, multiply the result by 4, getting A004018(10) = 8.
The total numbers of lattice points = 4r^2 = E(1) + ((E(2))/2 + ((E(3))/3 + ((E(4))/4 + ((E(5))/5 + .... Since E(even integers) are zero, E(integers of the form (4*k - 1)) = (-1), and E(integers of the form (4*k + 1)) = (+1); we are left with 4r^2 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ..., which is approximately equal to Pi(r^2). (End)
T(n,k) is also the number of parts in the partition of n into k equal parts. - Omar E. Pol, May 05 2020

			T(8,4) = 4 since 4 divides 8.
T(9,3) = 3 since 3 divides 9.
First few rows of the triangle:
  1;
  1, 2;
  1, 0, 3;
  1, 2, 0, 4;
  1, 0, 0, 0, 5;
  1, 2, 3, 0, 0, 6;
  1, 0, 0, 0, 0, 0, 7;
  1, 2, 0, 4, 0, 0, 0, 8;
  1, 0, 3, 0, 0, 0, 0, 0, 9;
  ...

David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, 2005, appendix.
L. Euler, "Discovery of a Most Extraordinary Law of the Numbers Concerning the Sum of Their Divisors"; pp. 358-367 of Robert M. Young, "Excursions in Calculus, An Interplay of the Continuous and the Discrete", MAA, 1992. See p. 366.

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Grant Sanderson, Pi hiding in prime regularities
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.
Leonhard Euler, Observatio de summis divisorum
Eric Weisstein's World of Mathematics, Divisor

Reversal = A127094
Cf. A127094, A123229, A127096, A127097, A127098, A127099, A000203, A126988, A127013, A127057, A038040, A024916, A060640, A001001.
Cf. A000005, A060640.
Cf. A027750.
Cf. A000012 (the first column), A020639, A059841 (the second column when multiplied by 2), A079978 (the third column when multiplied by 2), A079998 (the fifth column when multiplied by 5), A121262 (the fourth column when multiplied by 4).
Cf. A002654, A004018, A008683, A050873.

mod(row()-1;column()) - mod(row();column()) + 1 - Mats Granvik, Aug 31 2007

a127093 n k = a127093_row n !! (k-1)
a127093_row n = zipWith (*) [1..n] $ map ((0 ^) . (mod n)) [1..n]
a127093_tabl = map a127093_row [1..]
-- Reinhard Zumkeller, Jan 15 2011

A127093:=proc(n,k) if type(n/k, integer)=true then k else 0 fi end:
for n from 1 to 16 do seq(A127093(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Jan 20 2007

t[n_, k_] := k*Boole[Divisible[n, k]]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)
Table[ SeriesCoefficient[k*x^k/(1 - x^k), {x, 0, n}], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 14 2015 *)

trianglerows(n) = for(x=1, n, for(k=1, x, if(x%k==0, print1(k, ", "), print1("0, "))); print(""))
/* Print initial 9 rows of triangle as follows: */
trianglerows(9) \\ Felix Fröhlich, Mar 26 2019

k-th column is composed of "k" interspersed with (k-1) zeros.
Let M = A127093 as an infinite lower triangular matrix and V = the harmonic series as a vector: [1/1, 1/2, 1/3, ...]. then M*V = d(n), A000005: [1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...]. M^2 * V = A060640: [1, 5, 7, 17, 11, 35, 15, 49, 34, 55, ...]. - Gary W. Adamson, May 10 2007
T(n,k) = ((n-1) mod k) - (n mod k) + 1 (1 <= k <= n). - Mats Granvik, Aug 31 2007
T(n,k) = k * 0^(n mod k). - Reinhard Zumkeller, Jan 15 2011
G.f.: Sum_{k>=1} k * x^k * y^k/(1-x^k) = Sum_{m>=1} x^m * y/(1 - x^m*y)^2. - Robert Israel, Aug 08 2016
T(n,k) = Sum_{d|k} mu(k/d)*sigma(gcd(n,d)). - Ridouane Oudra, Apr 05 2025

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

A047201 Numbers not divisible by 5.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87

Offset: 1

N. J. A. Sloane

Original name was: Numbers that are congruent to {1, 2, 3, 4} mod 5.
More generally the sequence of numbers not divisible by some fixed integer m>=2 is given by a(n,m) = n-1+floor((n+m-2)/(m-1)). - Benoit Cloitre, Jul 11 2009
Complement of A008587. - Reinhard Zumkeller, Nov 30 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A019692, A045572, A179290.

a047201 n = a047201_list !! (n-1)
a047201_list = [x | x <- [1..], mod x 5 > 0]
-- Reinhard Zumkeller, Dec 17 2011

[Floor((15*n-1)/12): n in [1..70]]; // Vincenzo Librandi, Apr 06 2015

seq(floor((15*n-1)/12), n=1..56); # Gary Detlefs, Mar 07 2010

Select[Table[n,{n,200}],Mod[#,5]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)

a(n)= n+(n-1)\4 \\ corrected by Michel Marcus, Sep 02 2022

a(n)=n-1+floor((n+3)/4) \\ Benoit Cloitre, Jul 11 2009

def A047201(n): return n+(n-1>>2) # Chai Wah Wu, Feb 24 2025

[i for i in range(72) if gcd(5,i) == 1] # Zerinvary Lajos, Apr 21 2009

G.f.: (x+2*x^2+3*x^3+4*x^4+4*x^5+3*x^6+2*x^7+x^8)/(1-x^4)^2 (not reduced). - Len Smiley
a(n) = 5+a(n-4).
G.f.: x*(1+x+x^2+x^3+x^4)/((1-x)*(1-x^4)).
a(n) = n-1+floor((n+3)/4). - Benoit Cloitre, Jul 11 2009
A011558(a(n))=1; A079998(a(n))=0. - Reinhard Zumkeller, Nov 30 2009
a(n) = floor((15*n-1)/12). - Gary Detlefs, Mar 07 2010
a(n) = A225496(n) for n <= 42. - Reinhard Zumkeller, May 09 2013
From Wesley Ivan Hurt, Jun 22 2015: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5), n>5.
a(n) = (10*n-5-(-1)^n+2*(-1)^((2*n+5-(-1)^n)/4))/8. (End)
E.g.f.: 1 + (1/4)*(-cos(x) + (-3 + 5*x)*cosh(x) + sin(x) + (-2 + 5*x)*sinh(x)). - Stefano Spezia, Dec 01 2019
a(n) = floor((5*n-1)/4). - Wolfdieter Lang, Sep 30 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2-2/sqrt(5))*Pi/5 = A179290 * A019692 / 10. - Amiram Eldar, Dec 07 2021

Comment from Lekraj Beedassy, Dec 17 2006 is now the current name. - Wesley Ivan Hurt, Jun 25 2015

A011558 Expansion of (x + x^3)/(1 + x + ... + x^4) mod 2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Multiplicative with a(5^e) = 0, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005
Characteristic function of numbers coprime to 5. - Reinhard Zumkeller, Nov 30 2009
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the principal Dirichlet character mod 5. (The other real character mod 5 is A080891.)
Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 = 1.5791367... = (psi'(1/5) + psi'(2/5) + psi'(3/5) + psi'(4/5))/25 or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.192440... = -(psi''(1/5) + psi''(2/5) + psi''(3/5) + psi''(4/5))/250, where psi' and psi'' are the trigamma and tetragamma functions. (End)
a(n) is for n >= 1 also the characteristic function for rational g-adic integers (+n/5)A047201).%20See%20the%20definition%20in%20the%20Mahler%20reference,%20p.%207%20and%20also%20p.%2010.%20-%20_Wolfdieter%20Lang">g and also (-n/5)_g for all integers g >= 2 without a factor of 5 (A047201). See the definition in the Mahler reference, p. 7 and also p. 10. - _Wolfdieter Lang, Jul 11 2014
Conjecture: a(n+1) is the number of ways of partitioning n into distinct parts of A084215. - R. J. Mathar, Mar 01 2023

			G.f. = x + x^2 + x^3 + x^4 + x^6 + x^7 + x^8 + x^9 + x^11 + x^12 + ...

Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10).
K. Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Gilleland, Some Self-Similar Integer Sequences
R. Gold, Characteristic linear sequences and their coset functions, J. SIAM Applied. Math., 14 (1966), 980-985.
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A000035, A011655, A109720 coprimality with 2, 3, 7, respectively.
Cf. A168185, A145568, A168184, A168182, A168181, A097325, A166486.
Cf. A080891, A100047.

seq(n&^4 mod 5, n=0..50); # Gary Detlefs, Mar 20 2010

Mod[#,2]&/@CoefficientList[Series[(x+x^3)/(1+x+x^2+x^3+x^4) ,{x,0,100}], x] (* or *) Flatten[Table[{0,1,1,1,1},{30}]] (* Harvey P. Dale, May 15 2011 *)
a[ n_] := Sign@Mod[ n, 5]; (* Michael Somos, May 24 2015 *)

a(n)=!!(n%5) \\ Charles R Greathouse IV, Sep 23 2012

{a(n) = n%5>0}; /* Michael Somos, May 24 2015 */

(define (A011558 n) (if (zero? (modulo n 5)) 0 1)) ;; Antti Karttunen, Dec 21 2017

O.g.f.: x*(1+x+x^2+x^3)/(1-x^5). - Wolfdieter Lang, Feb 05 2009
From Reinhard Zumkeller, Nov 30 2009: (Start)
a(n) = 1 - A079998(n).
a(A047201(n))=1, a(A008587(n))=0.
A033437(n) = Sum_{k=0..n} a(k)*(n-k). (End)
a(n) = n^4 mod 5. - Gary Detlefs, Mar 20 2010
Sum_{n>=1} a(n)/n^s = L(s,chi) = (1-1/5^s)*Riemann_zeta(s), s > 1. - R. J. Mathar, Jul 31 2010
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) = sgn(n mod 5). - Wesley Ivan Hurt, Jun 30 2013
Euler transform of length 5 sequence [ 1, 0, 0, -1, 1]. - Michael Somos, May 24 2015
Moebius transform is length 5 sequence [ 1, 0, 0, 0, -1]. - Michael Somos, May 24 2015
G.f.: f(x) - f(x^5) where f(x) := x / (1 - x). - Michael Somos, May 24 2015
|a(n)| = |A080891(n)| = |A100047(n)|. - Michael Somos, May 24 2015

More terms from Antti Karttunen, Dec 21 2017

A079979 Characteristic function of multiples of six.

Original entry on oeis.org

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

Offset: 0

Vladimir Baltic, Feb 17 2003

Period 6: repeat [1, 0, 0, 0, 0, 0].
a(n)=1 if n=6k, a(n)=0 otherwise.
Decimal expansion of 1/999999.
Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,-1,0,1,2}.
Also, number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,1,2,3,4}.
a(n) is also the number of partitions of n such that each part is six (a(0)=1 because the empty partition has no parts to test equality with six). Hence a(n) is also the number of 2-regular graphs on n vertices with each part having girth exactly six. - Jason Kimberley, Oct 10 2011
This sequence is the Euler transformation of A185016. - Jason Kimberley, Oct 10 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..65538
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
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).
Index entries for characteristic functions

Cf. A002524-A002529, A010875, A072827, A072850-A072856, A079955-A080014, A097325, A122841.

Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), this sequence (g=6), A082784 (g=7).

Magma
```
&cat[[1,0^^5]^^30];
```

A079979 := func; [A079979:n in [0..59]];  // Jason Kimberley, Oct 10 2011

PadRight[{},120,{1,0,0,0,0,0}] (* Harvey P. Dale, Feb 19 2013 *)

a(n)=!(n%6) \\ Charles R Greathouse IV, Oct 10 2011

(define (A079979 n) (if (zero? (modulo n 6)) 1 0)) ;; Antti Karttunen, Dec 22 2017

a(n) = a(n-6).
G.f.: 1/(1-x^6).
a(n) = floor((1/2)*cos(n*Pi/3) + 1/2). - Gary Detlefs, May 16 2011
a(n) = floor(n/6) - floor((n-1)/6). - Tani Akinari, Oct 23 2012
a(n) = (((((v^n - w^n)^2)*(2 - (-1)^n)*(w^(2*n) + w^n - 3))^2 - 144)^2)/20736, where w = (-1+i*sqrt(3))/2, v = (1+i*sqrt(3))/2. - Bogart B. Strauss, Sep 20 2013
E.g.f.: (2*cos(sqrt(3)*x/2)*cosh(x/2) + cosh(x))/3. - Vaclav Kotesovec, Feb 15 2015

More terms from Antti Karttunen, Dec 22 2017

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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A008587 Multiples of 5: a(n) = 5 * n.

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A045572 Numbers that are odd but not divisible by 5.

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