A174542 Partial sums of odd Fibonacci numbers (A014437).

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

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Nov 28 2010

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000045, A099919, A014437.

Programs

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

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 *)
Accumulate[Select[Fibonacci[Range[50]],OddQ]] (* or *) LinearRecurrence[{1,4,-4,1,-1},{1,2,5,10,23},40] (* Harvey P. Dale, Sep 05 2023 *)

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

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)