A104537 Expansion of g.f.: (1+x)/(1+2*x+4x^2).

1, -1, -2, 8, -8, -16, 64, -64, -128, 512, -512, -1024, 4096, -4096, -8192, 32768, -32768, -65536, 262144, -262144, -524288, 2097152, -2097152, -4194304, 16777216, -16777216, -33554432, 134217728, -134217728, -268435456, 1073741824, -1073741824, -2147483648, 8589934592

Offset: 0

Author

Paul Barry, Mar 13 2005

Keywords

• easy
• sign

Comments

a(n+1) is the Hankel transform of C(2n,n)-C(2n+2,n+1). - Paul Barry, Mar 14 2008

a(n+1) is the Hankel transform of C(2n,n)-2*C(n)=((n-1)/(n+1))*C(2n,n), where C(n)=A000108(n). - Paul Barry, Mar 14 2008

Links

• Index entries for linear recurrences with constant coefficients, signature (-2,-4).

Crossrefs

Cf. A000108, A098158.

Programs

Magma

I:=[1, -1]; [n le 2 select I[n] else -2*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 16 2014

Maple

A104537:=n->2^n*cos(2*Pi*n/3): seq(A104537(n), n=0..40); # Wesley Ivan Hurt, Nov 16 2014

Mathematica

CoefficientList[Series[(1 + x) / (1 + 2 x + 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 16 2014 *)
LinearRecurrence[{-2,-4},{1,-1},40] (* Harvey P. Dale, Dec 02 2019 *)

Formula

G.f.: (1+x)/(1+2*x+4x^2).

a(n) = -2*a(n-1) - 4*a(n-2).

a(n) = 2^n*cos(2*Pi*n/3).

a(n) = Sum_{k=0..n} A098158(n,k)*(-1)^k*3^(n-k). - Philippe Deléham, Nov 14 2008

a(n) = (3^n/2^n)*Product_{i=1..n} (1/3 - tan((i-1/2)*Pi/(2*n))^2). - Gerry Martens, May 26 2011

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

a(n) = b(n) + b(n-1) where b(n) = 2^n*A049347(n). - R. J. Mathar, May 21 2019

Sum_{n>=0} 1/a(n) = -4/7. - Amiram Eldar, Feb 14 2023