A094707 Partial sums of repeated Fibonacci sequence.

0, 0, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 32, 40, 53, 66, 87, 108, 142, 176, 231, 286, 375, 464, 608, 752, 985, 1218, 1595, 1972, 2582, 3192, 4179, 5166, 6763, 8360, 10944, 13528, 17709, 21890, 28655, 35420, 46366, 57312, 75023, 92734, 121391, 150048, 196416

Offset

0,4

Comments

Equals row sums of triangle A139147 starting with "1". - Gary W. Adamson, Apr 11 2008

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

G.f. : x^2*(1+x)/((1-x)*(1-x^2-x^4)).

a(n) = a(n-1) + a(n+2) - a(n-3) + a(n-4) - a(n-5).

a(n) = Sum_{k=0..n} Fibonacci(floor(k/2)).

a(n) = -2 - (sqrt(5)/2 - 1/2)^(n/2)*((2*sqrt(5)/5 - 1)*cos(Pi*n/2) + sqrt(4*sqrt(5)/5 - 8/5)*sin(Pi*n/2)) - (sqrt(5)/2 + 1/2)^(n/2)*((sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5 - 1/2)*(-1)^n - sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5-1/2).

a(n) = A131524(n) + A131524(n+1). - R. J. Mathar, Jul 07 2011

a(n) = Fibonacci(n/2 +3) - 2 if n even, otherwise a(n) = 2*Fibonacci((n-1)/2 + 2) - 2. - G. C. Greubel, Feb 12 2023

Mathematica

LinearRecurrence[{1, 1, -1, 1, -1}, {0, 0, 1, 2, 3}, 50] (* Jean-François Alcover, Nov 18 2017 *)

Prog

(Magma) [Fibonacci(Floor((n+6)/2))*((n+1) mod 2) + 2*Fibonacci(Floor((n+3)/2))*(n mod 2) - 2: n in [0..60]]; // G. C. Greubel, Feb 12 2023
(SageMath)
def A094707(n): return fibonacci((n+6)//2) - 2 if (n%2==0) else 2*fibonacci((n+3)//2) - 2
[A094707(n) for n in range(61)] # G. C. Greubel, Feb 12 2023

Crossrefs

Cf. A000045, A103609, A131524, A139147.

Sequence in context: A038348 A239467 A035945 * A353864 A303663 A117995

Adjacent sequences: A094704 A094705 A094706 * A094708 A094709 A094710

Keyword

easy,nonn

Author

Paul Barry, May 21 2004

Status

approved