OFFSET
0,3
COMMENTS
Inverse binomial transform of (1,1,5,5,25,25,.....).
The absolute values are the constant terms of the reduction by x^2->x+1 of the polynomial p(n,x) given for d=sqrt(x+1) by p(n,x)=((x+d)^n-(x-d)^n)/(2d), for n>=1. The coefficient of x under this reduction is given by A103435. See A192232 for a discussion of reduction. - Clark Kimberling, Jun 29 2011
LINKS
FORMULA
G.f.: (1+2*x)/((1+(1+sqrt(5))*x)(1+(1-sqrt(5))*x)) = ( -1-2*x ) / ( -1-2*x+4*x^2 ).
E.g.f.: exp(-x)*(cosh(sqrt(5)*x)+sinh(sqrt(5)*x)/sqrt(5)).
a(n)=(sqrt(5)-1)^n*(sqrt(5)/10+1/2)+(-sqrt(5)-1)^n*(1/2-sqrt(5)/10).
(-1)^n*a(n) = sum(k=0..n, binomial(n,k)*(F(n+1)-F(n))), F(n) Fibonacci number A000045. - Peter Luschny, Oct 01 2014
a(n) = (-2)^n *A000045(n-1). - Robert Israel, Oct 02 2014
MAPLE
seq((-2)^n * combinat:-fibonacci(n-1), n = 0 .. 100); # Robert Israel, Oct 02 2014
MATHEMATICA
LinearRecurrence[{-2, 4}, {1, 0}, 40] (* Harvey P. Dale, Oct 10 2018 *)
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Jul 17 2003
STATUS
approved