OFFSET
0,1
COMMENTS
G.f. is inverse of cyclotomic(4,x). Unsigned: A000035(n+1).
Real part of i^n and imaginary part of i^(n+1), i=sqrt(-1). - Reinhard Zumkeller, Jul 22 2007
The BINOMIAL transform generates A009116(n); the inverse BINOMIAL transform generates (-1)^n*A009116(n). - R. J. Mathar, Apr 07 2008
a(n-1), n >= 1, is the nontrivial Dirichlet character modulo 4, called Chi_2(4;n) (the trivial one is Chi_1(4;n) given by periodic(1,0) = A000035(n)). See the Apostol reference, p. 139, the k = 4, phi(k) = 2 table. - Wolfdieter Lang, Jun 21 2011
a(n-1), n >= 1, is the character of the Dirichlet beta function. - Daniel Forgues, Sep 15 2012
a(n-1), n >= 1, is also the (strongly) multiplicative function h(n) of Theorem 5.12, p. 150, of the Niven-Zuckerman reference. See the formula section. This function h(n) can be employed to count the integer solutions to n = x^2 + y^2. See A002654 for a comment with the formula. - Wolfdieter Lang, Apr 19 2013
This sequence is duplicated in A101455 but with offset 1. - Gary Detlefs, Oct 04 2013
For n >= 2 this gives the determinant of the bipartite graph with 2*n nodes and the adjacency matrix A(n) with elements A(n;1,2) = 1 = A(n;n,n-1), and for 1 < i < n A(n;i,i+1) = 1 = A(n;i,i-1), otherwise 0. - Wolfdieter Lang, Jun 25 2023
REFERENCES
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980), p. 150.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:1 at page 300.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry and Nikolaos Pantelidis, On pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays, J Algebr Comb 54, 399-423 (2021).
PeakMath, L-functions and the Langlands program (RH Saga S1E2), YouTube video (2024).
Eric Weisstein's World of Mathematics, Kronecker Symbol.
Wikipedia, Dirichlet beta function.
Wikipedia, Kronecker Symbol.
Index entries for linear recurrences with constant coefficients, signature (0,-1).
FORMULA
G.f.: 1/(1+x^2).
E.g.f.: cos(x).
a(n) = (1/2)*((-i)^n + i^n), where i = sqrt(-1). - Mitch Harris, Apr 19 2005
a(n) = (1/2)*((-1)^(n+floor(n/2)) + (-1)^floor(n/2)).
Recurrence: a(n)=a(n-4), a(0)=1, a(1)=0, a(2)=-1, a(3)=0.
a(n) = T(n, 0) = A053120(n, 0); T(n, x) Chebyshev polynomials of the first kind. - Wolfdieter Lang, Aug 21 2009
a(n) = S(n, 0) = A049310(n, 0); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind.
Sum_{k>=0} a(k)/(k+1) = Pi/4. - Jaume Oliver Lafont, Mar 30 2010
a(n) = Sum_{k=0..n} A101950(n,k)*(-1)^k. - Philippe Deléham, Feb 10 2012
a(n) = (1/2)*(1 + (-1)^n)*(-1)^(n/2). - Bruno Berselli, Mar 13 2012
a(0) = 1, a(n-1) = 0 if n is even, a(n-1) = Product_{j=1..m} (-1)^(e_j*(p_j-1)/2) if the odd n-1 = p_1^(e_1)*p_2^(e_2)*...*p_m^(e_m) with distinct odd primes p_j, j=1..m. See the function h(n) of Theorem 5.12 of the Niven-Zuckerman reference. - Wolfdieter Lang, Apr 19 2013
a(n) = (-4/(n+1)), n >= 0, where (k/n) is the Kronecker symbol. See the Eric Weisstein and Wikipedia links. Thanks to Wesley Ivan Hurt. - Wolfdieter Lang, May 31 2013
a(n) = R(n,0)/2 with the row polynomials R of A127672. This follows from the product of the zeros of R, and the formula Product_{k=0..n-1} 2*cos((2*k+1)*Pi/(2*n)) = (1 + (-1)^n)*(-1)^(n/2), n >= 1 (see the Gradstein and Ryshik reference, p. 63, 1.396 4., with x = sqrt(-1)). - Wolfdieter Lang, Oct 21 2013
a(n) = Sum_{k=0..n} i^(k*(k+1)), where i=sqrt(-1). - Bruno Berselli, Mar 11 2015
Dirichlet g.f. of a(n) shifted right: L(chi_2(4),s) = beta(s) = (1-2^(-s))*(d.g.f. of A034947), see comments by Lang and Forgues. - Ralf Stephan, Mar 27 2015
a(n) = cos(3*n*Pi/2). - Ridouane Oudra, Sep 29 2024
EXAMPLE
With a(n-1) = h(n) of Niven-Zuckerman: a(62) = h(63) = h(3^2*7^1) = (-1)^(2*1)*(-1)^(1*3) = -1 = h(3)^2*h(7) = a(2)^2*a(6) = (-1)^2*(-1) = -1. - Wolfdieter Lang, Apr 19 2013
MAPLE
A056594 := n->(1-irem(n, 2))*(-1)^iquo(n, 2); # Peter Luschny, Jul 27 2011
MATHEMATICA
CoefficientList[Series[1/(1 + x^2), {x, 0, 50}], x]
a[n_]:= KroneckerSymbol[-4, n+1]; Table[a[n], {n, 0, 93}]. (* Thanks to Jean-François Alcover. - Wolfdieter Lang, May 31 2013 *)
CoefficientList[Series[1/Cyclotomic[4, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
PROG
(PARI) {a(n) = real( I^n )}
(PARI) {a(n) = kronecker(-4, n+1) }
(Magma) &cat[ [1, 0, -1, 0]: n in [0..23] ]; // Bruno Berselli, Feb 08 2011
(Maxima) A056594(n) := block(
[1, 0, -1, 0][1+mod(n, 4)]
)$ /* R. J. Mathar, Mar 19 2012 */
(Python)
def A056594(n): return (1, 0, -1, 0)[n&3] # Chai Wah Wu, Sep 23 2023
CROSSREFS
KEYWORD
sign,easy,changed
AUTHOR
Wolfdieter Lang, Aug 04 2000
STATUS
approved