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A174542 Partial sums of odd Fibonacci numbers (A014437). 1
1, 2, 5, 10, 23, 44, 99, 188, 421, 798, 1785, 3382, 7563, 14328, 32039, 60696, 135721, 257114, 574925, 1089154, 2435423, 4613732, 10316619, 19544084, 43701901, 82790070, 185124225, 350704366, 784198803, 1485607536, 3321919439, 6293134512, 14071876561 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

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

FORMULA

a(2n) = (1/2)*(Fibonacci(3n+2)-1).

a(2n+1) = (1/2)*(Fibonacci(3n+1)+Fibonacci(3n+3)-1).

a(n) = a(n-1)+4*a(n-2)-4*a(n-3)+a(n-4)-a(n-5) for n>4. - Colin Barker, Oct 26 2014

G.f.: x*(x^3-x^2+x+1) / ((x-1)*(x^4+4*x^2-1)). - Colin Barker, Oct 26 2014

From Vladimir Reshetnikov, Oct 30 2015: (Start)

a(n) = ((sin(Pi*n/2)*sqrt(5/phi) - cos(Pi*n/2)/phi^2)/phi^(3*n/2) + (sqrt(5*phi)*sin(Pi*n/2)^2 + phi^2*cos(Pi*n/2)^2)*phi^(3*n/2))/(2*sqrt(5)) - 1/2, where phi=(1+sqrt(5))/2.

E.g.f.: phi^2*cosh(phi^(3/2)*x)/(2*sqrt(5)) + sqrt(phi)*sinh(phi^(3/2)*x)/2 - cos(x/phi^(3/2))/(2*sqrt(5)*phi^2) + sin(x/phi^(3/2))/(2*sqrt(phi)) - exp(x)/2.

(End)

MATHEMATICA

s=0; lst={}; Do[f=Fibonacci[n]; If[OddQ[f], AppendTo[lst, s+=f]], {n, 0, 5!}]; lst

CoefficientList[Series[(x^3 - x^2 + x + 1)/((x - 1) (x^4 + 4 x^2 - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 27 2014 *)

PROG

(PARI) Vec(x*(x^3-x^2+x+1)/((x-1)*(x^4+4*x^2-1)) + O(x^100)) \\ Colin Barker, Oct 26 2014

(MAGMA) &cat[[(1/2)*(Fibonacci(3*n+2)-1), (1/2)*(Fibonacci(3*n+1)+Fibonacci(3*n+3)-1)]: n in [1..30]]; // Vincenzo Librandi, Oct 27 2014

CROSSREFS

Cf. A000045, A099919, A014437.

Sequence in context: A226455 A284139 A068054 * A007182 A191386 A323988

Adjacent sequences:  A174539 A174540 A174541 * A174543 A174544 A174545

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Nov 28 2010

STATUS

approved

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Last modified September 30 14:57 EDT 2020. Contains 337439 sequences. (Running on oeis4.)