A193356 - OEIS

%I #97 Jan 17 2024 09:12:24

%S 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,

%T 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,

%U 57,0,59,0,61,0,63,0,65,0,67,0,69,0,71,0,73,0,75

%N If n is even then 0, otherwise n.

%C 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

%C 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

%C Row sums of A057211. - _Omar E. Pol_, Mar 05 2014

%C Column k=2 of triangle A196020. - _Omar E. Pol_, Aug 07 2015

%C 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

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

%H Vincenzo Librandi, <a href="/A193356/b193356.txt">Table of n, a(n) for n = 1..1000</a>

%H C. Kravvaritis, <a href="http://dx.doi.org/10.2478/spma-2014-0019">Determinant evaluations for binary circulant matrices</a>, Special Matrices, V2(1) (2014), 187-199.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).

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

%F Dirichlet g.f.: (1-2^(1-s))*zeta(s-1). - _R. J. Mathar_, Aug 01 2011

%F G.f.: x*(1+x^2)/(1-x^2)^2. - _Philippe Deléham_, Feb 13 2012

%F a(n) = A027656(A042948(n-1)) = (1-(-1)^n)*n/2. - _Bruno Berselli_, Feb 19 2012

%F a(n) = n * (n mod 2). - _Wesley Ivan Hurt_, Jun 29 2013

%F G.f.: Sum_{n >= 1} A000010(n)*x^n/(1 + x^n). - _Mircea Merca_, Feb 22 2014

%F a(n) = 2*a(n-2)-a(n-4), for n>4. - _Wesley Ivan Hurt_, Aug 07 2015

%F E.g.f.: x*cosh(x). - _Robert Israel_, Feb 03 2016

%F 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

%F a(n) = Sum_{k=1..n} (-1)^((n-k)*k). - _Rick L. Shepherd_, Sep 18 2020

%F 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

%p A193356:=n->(1-(-1)^n)*n/2: seq(A193356(n), n=1..100); # _Wesley Ivan Hurt_, Aug 07 2015

%t Table[PowerMod[n,n,2*n], {n,200}]

%o (PARI) a(n)=if(n%2,n) \\ _Charles R Greathouse IV_, Jul 24 2011

%o (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

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

%K nonn,easy,mult

%O 1,3

%A _José María Grau Ribas_, Jul 24 2011

%E Formula for a(n) extended by _Wolfdieter Lang_, Dec 21 2011