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A049602 a(n) = (Fibonacci(2*n)-(-1)^n*Fibonacci(n))/2. 4
0, 1, 1, 5, 9, 30, 68, 195, 483, 1309, 3355, 8900, 23112, 60813, 158717, 416325, 1088661, 2852242, 7463884, 19546175, 51163695, 133962621, 350695511, 918170280, 2403740304, 6293172025, 16475579353, 43133883845, 112925557953 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A049602 gives the coefficients of x in the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1.  For the constant terms, see A192352.  For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. - Clark Kimberling, Jun 29 2011

LINKS

Table of n, a(n) for n=0..28.

Index entries for linear recurrences with constant coefficients, signature (2,3,-4,1).

FORMULA

a(n)=Sum{T(2i+1, n-2i-1): i=0, 1, ..., [ (n+1)/2 ]}, array T as in A049600.

Cosh transform of Fibonacci numbers A000045 (or mean of binomial and inverse binomial transforms of A000045). E.g.f.: cosh(x)(2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - Paul Barry, May 10 2003

a(n)=sum{k=0..floor(n/2), C(n, 2k)Fib(n-2k)}; - Paul Barry, May 01 2005

a(n)=2a(n-1)+3a(n-2)-4a(n-3)+a(n-4). - Paul Curtz, Jun 16 2008

G.f.: x(1-x)/((1+x-x^2)(1-3x+x^2)); a(n)=sum{k=0..n-1, (-1)^(n-k+1)*F(2k+2)*F(n-k+1)}; - Paul Barry, Jul 11 2008

MATHEMATICA

LinearRecurrence[{2, 3, -4, 1}, {0, 1, 1, 5}, 30] (* Harvey P. Dale, Jul 07 2017 *)

PROG

(PARI) a(n)=(fibonacci(2*n)-(-1)^n*fibonacci(n))/2 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A049601.

Sequence in context: A303988 A242329 A214902 * A119031 A271129 A271152

Adjacent sequences:  A049599 A049600 A049601 * A049603 A049604 A049605

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Simpler description from Vladeta Jovovic and Thomas Baruchel, Aug 24 2004

More terms from Paul Curtz, Jun 16 2008

STATUS

approved

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Last modified February 18 00:19 EST 2019. Contains 320237 sequences. (Running on oeis4.)