A079998 The characteristic function of the multiples of five.

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

Author

Vladimir Baltic, Feb 10 2003

Keywords

• nonn
• easy

Comments

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

References

• 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.

Links

• 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).

Crossrefs

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

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Scheme

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

Formula

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

Extensions

More terms from Antti Karttunen, Dec 21 2017