OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-16,8).
FORMULA
G.f.: (1-5*x+4*x^2)/(1-8*x+16*x^2-8*x^3) = (1-5*x+4*x^2)/((1-2*x)*(1-6*x+4*x^2)).
a(n) = 2^(n-1) + (3-sqrt(5))^n*((5-sqrt(5))/20) + (3+sqrt(5))^n*((5+sqrt(5))/20).
a(n) = 2^(n-1) + A082761(n)/2. - R. J. Mathar, Sep 30 2012
a(0)=1, a(1)=3, a(2)=12, a(n) = 8*a(n-1) - 16*a(n-2) + 8*a(n-3). - Harvey P. Dale, Aug 14 2013
a(n) = 2^(n-1)*(Fibonacci(2*n+1) + 1). - G. C. Greubel, Nov 24 2019
MAPLE
with(combinat); seq(2^(n-1)*(fibonacci(2*n+1) + 1), n=0..30); # G. C. Greubel, Nov 24 2019
MATHEMATICA
CoefficientList[Series[(1-5x+4x^2)/((1-2x)(1-6x+4x^2)), {x, 0, 40}], x] (* or *) LinearRecurrence[{8, -16, 8}, {1, 3, 12}, 40] (* Harvey P. Dale, Aug 14 2013 *)
PROG
(PARI) vector(31, n, 2^(n-2)*(fibonacci(2*n-1) + 1)) \\ G. C. Greubel, Nov 24 2019
(Magma) [2^(n-1)*(Fibonacci(2*n+1) + 1): n in [0..30]]; G. C. Greubel, Nov 24 2019
(Sage) [2^(n-1)*(fibonacci(2*n+1) + 1) for n in (0..30)] # G. C. Greubel, Nov 24 2019
(GAP) List([0..30], n-> 2^(n-1)*(Fibonacci(2*n+1) + 1)); # G. C. Greubel, Nov 24 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 14 2010
STATUS
approved