A056594 Period 4: repeat [1,0,-1,0]; expansion of 1/(1 + x^2).
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, -1, 0, 1, 0, -1, 0, 1, 0
Offset: 0
Examples
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
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).
- Index entries for sequences related to Chebyshev polynomials.
- Index to sequences related to inverse of cyclotomic polynomials.
Programs
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Magma
&cat[ [1, 0, -1, 0]: n in [0..23] ]; // Bruno Berselli, Feb 08 2011
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Maple
A056594 := n->(1-irem(n,2))*(-1)^iquo(n,2); # Peter Luschny, Jul 27 2011
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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 *) -
Maxima
A056594(n) := block( [1,0,-1,0][1+mod(n,4)] )$ /* R. J. Mathar, Mar 19 2012 */
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PARI
{a(n) = real( I^n )} -
PARI
{a(n) = kronecker(-4, n+1) } -
Python
def A056594(n): return (1,0,-1,0)[n&3] # Chai Wah Wu, Sep 23 2023
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
Comments