A193356 - OEIS

A193356

If n is even then 0, otherwise n.

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

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OFFSET

1,3

COMMENTS

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

REFERENCES

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

LINKS

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).

FORMULA

a(n) = n^k mod 2n, for any k>=2, also for k=n.

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

MAPLE

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}]

PROG

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

(Magma) 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

CROSSREFS

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

Sequence in context: A318517 A245494 A167465 * A347347 A071649 A341310